Mirror Symmetry, Autoequivalences,and Bridgeland Stability Conditions

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Citation Fan, Yu-Wei. 2019. Mirror Symmetry, Autoequivalences, andBridgeland Stability Conditions. Doctoral dissertation, HarvardUniversity, Graduate School of Arts & Sciences.

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Mirror Symmetry, Autoequivalences, and Bridgeland Stability Conditions

A dissertation presented

by

Yu-Wei Fan

to

The Department of Mathematics

in partial fulfillment of the requirements

for the degree of

Doctor of Philosophy

in the subject of

Mathematics

Harvard University

Cambridge, Massachusetts

April 2019

c© 2019 Yu-Wei Fan

All rights reserved.

Dissertation Advisor: Professor Shing-Tung Yau Yu-Wei Fan

Mirror Symmetry, Autoequivalences, and Bridgeland Stability Conditions

Abstract

The present thesis studies various aspects of Calabi–Yau manifolds, including mirror

symmetry, systolic geometry, and dynamical systems.

We construct the mirror operation of Atiyah flop in symplectic geometry. We construct

the mirror metric of the Weil–Petersson metric on the complex moduli space of Calabi–Yau

manifolds, in terms of derived categories and Bridgeland stability conditions.

We propose a new generalization of Loewner’s torus systolic inequality from the per-

spective of Calabi–Yau geometry, and prove a generalized systolic inequality for generic K3

surfaces.

We study the dynamical systems formed by autoequivalences on the derived categories of

Calabi–Yau manifolds, and find the first counterexamples of Kikuta–Takahashi’s conjecture

on categorical entropy.

iii

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

Introduction 1

1 Mirror of Atiyah flop 91.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2 Review on flops and Bridgeland stability conditions . . . . . . . . . . . . . . . 14

1.2.1 Bridgeland stability conditions and crepant small resolutions . . . . . 141.2.2 The conifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.3 Review on the SYZ mirror of the conifold . . . . . . . . . . . . . . . . . . . . . 161.4 A-flop in symplectic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.4.1 Mirror of Atiyah flop as a symplectomorphism . . . . . . . . . . . . . 201.4.2 Lagrangian fibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.4.3 A-flop on Lagrangian submanifolds . . . . . . . . . . . . . . . . . . . . 271.4.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.5 Derived Fukaya category of the deformed conifold . . . . . . . . . . . . . . . 341.5.1 Geometric objects of DbF . . . . . . . . . . . . . . . . . . . . . . . . . . 391.5.2 Mirror to perverse point sheaves . . . . . . . . . . . . . . . . . . . . . . 45

1.6 Non-commutative mirror functor for the deformed conifold and stabilityconditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461.6.1 Floer cohomology of L . . . . . . . . . . . . . . . . . . . . . . . . . . . 471.6.2 Construction of mirror from L . . . . . . . . . . . . . . . . . . . . . . . 511.6.3 Transformation of L0 and L1 and their central charges . . . . . . . . . 531.6.4 Stables on quiver side . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551.6.5 Transformation of special Lagrangians in DbF before/after flop . . . 581.6.6 Bridgeland stability on DbF . . . . . . . . . . . . . . . . . . . . . . . . 64

2 Mirror of Weil–Petersson metric 662.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662.2 Bridgeland stability conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

iv

2.2.1 Review: Bridgeland stability conditions . . . . . . . . . . . . . . . . . . 692.2.2 Central charge via twisted Mukai pairing . . . . . . . . . . . . . . . . . 71

2.3 Weil–Petersson geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742.3.1 Classical Weil–Petersson geometry . . . . . . . . . . . . . . . . . . . . . 742.3.2 Weil–Petersson geometry on StabN (D) . . . . . . . . . . . . . . . . . . 752.3.3 Refining conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

2.4 Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 782.4.1 Self-product of elliptic curve . . . . . . . . . . . . . . . . . . . . . . . . 792.4.2 Split abelian surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 862.4.3 Abelian variety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 872.4.4 Quintic threefold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

3 Systolic inequality on K3 surfaces 913.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 913.2 Categorical systole and categorical volume . . . . . . . . . . . . . . . . . . . . 97

3.2.1 Categorical systole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 973.2.2 Categorical volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

3.3 Elliptic curves case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1003.4 K3 surfaces case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1023.5 Further studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

3.5.1 Optimal systolic ratio of K3 surfaces of Picard rank one . . . . . . . . 1093.5.2 Systolic inequality for K3 surfaces of higher Picard rank . . . . . . . . 1103.5.3 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

4 Entropy of autoequivalences 1124.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1124.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

4.2.1 Categorical entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1154.2.2 Spherical objects and spherical twists . . . . . . . . . . . . . . . . . . . 116

4.3 Computation of categorical entropy . . . . . . . . . . . . . . . . . . . . . . . . 1164.4 Counterexample of Kikuta-Takahashi . . . . . . . . . . . . . . . . . . . . . . . 1204.5 Categorical entropy of P-twists . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

References 125

v

List of Tables

3.1 Correspondence among flat surfaces, holomorphic top forms on Calabi–Yaumanifolds, and stability conditions on triangulated categories. . . . . . . . . . 94

vi

List of Figures

1.1 The Atiyah flop. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2 The local conifold is self-mirror. More precisely for the local conifold, a

resolution and its flop are equivalent, so the upper hemisphere should beidentified with the lower hemisphere. . . . . . . . . . . . . . . . . . . . . . . . 13

1.3 The base and discriminant loci of Lagrangian fibrations in conifold transition. 171.4 Lagrangian S3 seen from the double conic fibration. . . . . . . . . . . . . . . . 201.5 The symplectic reduction of Lagrangian torus fibers before and after the A-flop. 261.6 The discriminant locus before and after the A-flop. . . . . . . . . . . . . . . . 291.7 A-flop shown in the double conic fibration picture. . . . . . . . . . . . . . . . 291.8 Polytopes in the polyhedral decomposition of the affine base of Schoen’s CY. 331.9 Sequence of Lagrangian spheres in Xs=0. . . . . . . . . . . . . . . . . . . . . . 351.10 Special Lagrangian spheres S′k in Xs=1. . . . . . . . . . . . . . . . . . . . . . . . 361.11 Transformation of L0 and L1 by the symplectomorphism ρ. . . . . . . . . . . 371.12 Perturbation of Lc, Li and polygons contributing to mb

1 on CF(L, (Lc, ρ)). . . . 401.13 Two triangles contributing to m1 and m2. . . . . . . . . . . . . . . . . . . . . . 411.14 Triangles contributing to m1 and m2 on CF(Sm[1]⊕Sm+1, Lc) and CF(Lc, Sm[1]⊕

Sm+1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431.15 Transformation of torus fibers under ρ. . . . . . . . . . . . . . . . . . . . . . . 451.16 A trajectory of the Morse function fi from S1

a to S1b. . . . . . . . . . . . . . . . 49

1.17 Pearl trajectory contributing to m3(X, Y, Z). . . . . . . . . . . . . . . . . . . . . 501.18 Morse trees for m3(X, Y, Z). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511.19 Contributing holomorphic polygons to (the differential of) Ψ(Sm). . . . . . . 63

vii

Acknowledgments

First and foremost, I want to express my greatest gratitude to my advisor Shing-Tung Yau

for his constant encouragement, insightful ideas and tremendous support. Without him this

work would not be possible.

It is my great pleasure to thank my collaborators Hansol Hong, Atsushi Kanazawa and

Siu-Cheong Lau for fruitful collaboration and many inspiring discussions. I thank partic-

ularly Atsushi Kanazawa for encouraging me to work on Bridgeland stability conditions,

which later became one of the main themes of my thesis.

I want to express my sincere gratitude to Simion Filip, Fabian Haiden, Ludmil Katzarkov,

Yu-Shen Lin, Emanuele Macrì and Yukinobu Toda for teaching me lots of mathematics and

sharing ideas with me. I also want to thank Lino Amorim, Mandy Cheung, Tristan Collins,

Philip Engel, Qingyuan Jiang, Kuan-Wen Lai, Conan Leung, Hseuh-Yung Lin, Chien-

Hao Liu, Chiu-Chu Melissa Liu, Yijia Liu, Cheuk Yu Mak, Genki Ouchi, Alexander Perry,

Yu Qiu, Shu-Heng Shao, Arnav Tripathy, Chien-Hsun Wang, Dan Xie, Jingyu Zhao and

Jie Zhou for many helpful and interesting discussions.

I am grateful to Chin-Lung Wang and Hui-Wen Lin for introducing mirror symmetry

to me when I was an undergraduate at National Taiwan University. I also am grateful to

Mboyo Esole, Yuan-Pin Lee and Pei-Yu Tsai for offering very helpful career advice to me.

I thank all the Yau’s seminar participants including Teng Fei, Peter Smillie, Chenglong Yu,

Netanel Rubin-Blaier, Charles Doran, Peng Gao, An Huang, Yoosik Kim, Tsung-Ju Lee,

Hossein Movasati, Sze-Man Ngai, Valentino Tosatti, Yi Xie, Mei-Heng Yueh, Boyu Zhang,

Dingxin Zhang, Xin Zhou, Yang Zhou and Jonathan Zhu for so many interesting talks and

conversations that broaden the scope of my interest.

I want to thank my fellow graduate students in the True Alcove: Chi-Yun Hsu, Koji Shimizu,

Yunqing Tang and Cheng-Chiang Tsai, as well as my friends Hung-Yu Chien, Chieh-

Hsuan Kao and Ya-Ling Kao for their friendship and support.

At the end, I want to express my sincere gratitude to my parents Chin-Hung Fan and

Li-Chun Tseng, and my sister Yu-Chen Fan for their love and support. Last but not least, I

viii

want to express my deepest gratitude to my love Xueyin Shao, who supported me during

my most difficult times. I want to take this chance to say I love you.

ix

Introduction

Calabi–Yau manifolds have very rich geometric structures and have been studied ex-

tensively in mathematics and physics. The present thesis studies three different aspects of

Calabi–Yau manifolds: mirror symmetry, systolic geometry, and dynamical systems.

Mirror symmetry was first introduced by string theorists. It predicts that Calabi–Yau

manifolds come in pairs, in which the complex geometry of one is equivalent to the symplectic

geometry of the other, and vice versa. One mathematical formulation of this prediction was

proposed by Kontsevich [65].

Conjecture 0.0.1 (Homological mirror symmetry conjecture [65]). Let X be a Calabi–Yau

manifold. There exists a Calabi–Yau manifold Y such that there are equivalences between triangulated

categories

DbCoh(X) ∼= DπFuk(Y) and DbCoh(Y) ∼= DπFuk(X).

Here DbCoh(X) and DπFuk(X) denote the derived category of coherent sheaves and the derived

Fukaya category of X, respectively.

Roughly speaking, the objects in the derived category DbCoh(X) are complexes of

holomorphic vector bundles on X, and the objects in the derived Fukaya category DπFuk(X)

are Lagrangian submanifolds in X with certain extra data. Homological mirror symmetry

conjecture has been proved in many cases, see for instance [82, 88, 90].

Mirror symmetry phenomenon naturally leads to the following question.

Question 0.0.2. Given an “object" (e.g. operation, geometric structure, etc.) in complex geometry,

can one construct its counterpart in symplectic geometry, or vice versa?

1

Chapter 1 and Chapter 2 contain results along this line of thought.

Chapter 1: Mirror of Atiyah flop

Flops are fundamental operations in birational geometry. Among them, Atiyah flop

is the simplest and the most well-known one. An Atiyah flop X → X ← X† contracts

a (−1,−1)-rational curve C in a complex threefold X and resolve the resulting conifold

singularity with another (−1,−1)-rational curve C†.

The goal of Chapter 1 is to construct the mirror of Atiyah flop in symplectic geometry

under mirror symmetry.

Theorem 0.0.3 (= Proposition 1.1.1). Given a symplectic sixfold (Y, ω) and a Lagrangian three-

sphere S ⊂ Y, we construct another symplectic sixfold (Y†, ω†) with a corresponding Lagrangian

three-sphere S† ⊂ Y†, together with a symplectomorphism f (Y,S) : (Y, ω) → (Y†, ω†). It has the

property that f (Y†,S†) f (Y,S) = τ−1

S , where τS is the Dehn twist along the Lagrangian sphere S.

The symplectomorphism f (Y,S) in Theorem 0.0.3 is the mirror of Atiyah flop. The

contracted (−1,−1)-rational curve in algebraic geometry corresponds to the Lagrangian

three-sphere in symplectic geometry. Bridgeland [13] shows that threefolds related by a flop

are derived equivalent: Db(X) ∼= Db(X†). The property f (Y†,S†) f (Y,S) = τ−1

S is mirror to

the fact that the composition of two flop functors Db(X)→ Db(X†)→ Db(X) is the inverse

of the Seidel–Thomas spherical twist [89] by OC(−1).

Let DX/X ⊂ Db(X) be the subcategory which consists of objects supported on C.

This subcategory captures the local geometry of the flopping curve. Then Bridgeland’s

equivalence restricts to an equivalence DX/X∼= DX†/X. It is proved by Chan–Pomerleano–

Ueda [21] that DX/X is equivalent to certain derived Fukaya category DbFY. We prove the

following compatibility result between the Atiyah flop and our mirror Atiyah flop.

Theorem 0.0.4 (= Proposition 1.5.4). The symplectomorphism f (Y,S) induces an equivalence

between the derived Fukaya categories DbFY∼= DbFY† . Moreover, the equivalence is the same as the

composition DbFY∼= DX/X

∼= DX†/X∼= DbFY† , where the first and third equivalences are given

by Chan–Pomerleano–Ueda [21], and the second equivalence is Bridgeland’s flopping equivalence.

2

Note that unlike the Atiyah flop which produces different complex manifold, its mirror

f (Y,S) is a symplectomorphism. It is not very surprising since symplectic geometry is much

softer than complex geometry. On the other hand, we can endow more structures so that

the effect of mirror Atiyah flop can be seen. One way to do so is to consider the Bridgeland

stability conditions [14] on DbFY.

Theorem 0.0.5 (= Theorem 1.1.3). Let Y = u1v1 = z + q, u2v2 = z + 1, z 6= 0 be the deformed

conifold and ΩY = dz∧ du1 ∧ du2 be a holomorphic volume form on Y. Then there exists a collection

P of graded special Lagrangian submanifolds which defines a geometric stability condition (Z ,P)

on DbFY. Moreover, the mirror Atiyah flop f (Y,S) defines another geometric stability condition

(Z†,P†) with respect to ( f (Y,S))∗ΩY† . Finally, (Z ,P) and (Z†,P†) are related by a wall-crossing

in the space of Bridgeland stability conditions Stab(DbFY), which matches with Toda’s wall-crossing

in Stab(DX/X) on the mirror [98].

Another way to see the effect of mirror Atiyah flop is by equipping the symplectic sixfold

Y with a Lagrangian fibration. See Theorem 1.1.2 for more details.

Chapter 2: Mirror of Weil–Petersson metric

The moduli space of complex structures Mcpx(Y) on a Calabi–Yau manifold Y has a

canonical Kähler metric, the Weil–Petersson metric. The existence of such a natural metric

often implies strong results that one can not obtain by purely algebraic methods. In the case

of Calabi–Yau threefold, this metric provides a fundamental differential geometric tool, the

special Kähler geometry, to study mirror symmetry.

The goal of Chapter 2 is to construct the mirror object of the Weil–Petersson metric.

Under mirror symmetry,Mcpx(Y) should be identified with the so-called “stringy Kähler

moduli space"MKah(X) of a mirror Calabi–Yau manifold X. When dim(X) ≤ 2,MKah(X)

is well-defined via Bridgeland stability conditions by a work of Bayer–Bridgeland [8].

When dim(X) ≥ 3, it is conjectured by Bridgeland [14, 16] that there is an embedding

MKah(X) → Aut(Db(X))\Stab(Db(X))/C. In other words, the stringy Kähler moduli

space is encoded in the space of Bridgeland stability conditions on the derived category

3

Db(X). Our strategy therefore is to first define the Weil–Petersson geometry on the space of

Bridgeland stability conditions, then restrict to the stringy Kähler moduli space.

Definition 0.0.6 (= Definition 2.3.3). For any Calabi–Yau categoryD, we define the Weil–Petersson

metric on Stab+(D)/C for an appropriate subset Stab+(D) ⊂ Stab(D). The metric descends to

the double quotient space Aut(D)\Stab+(D)/C.

We compute several low-dimensional examples to justify our definition of Weil–Petersson

metric:

Theorem 0.0.7 (= Example 2.3.5 and Theorem 2.1.1). We compute the following examples of

Weil–Petersson metric on the space of Bridgeland stability conditions.

• Let E be an elliptic curve. ThenMKah(E) ∼= Aut(Db(E))\Stab+(Db(E))/C ∼= PSL(2, Z)\H;

our Weil–Petersson metric coincides with the Poincaré metric on H.

• Let A = Eτ × Eτ be the self-product of a generic elliptic curve. Then MKah(A) ∼=

AutCY(Db(A))\Stab+(Db(A))/C ∼= Sp(4, Z)\H2 is the Siegel modular variety; our Weil–

Petersson metric coincides with the Bergman metric on Sp(4, Z)\H2.

The Weil–Petersson metric on Aut(D)\Stab+(D)/C is a degenerate metric in general.

However, the Weil–Petersson metric on the complex moduli Mcpx(Y) is non-degenerate.

Hence we expect the non-degeneracy condition can be used to characterize the stringy

Kähler moduli spaceMKah(X).

Conjecture 0.0.8 (= Conjecture 2.3.6). For dim(X) ≥ 3, there exists an embedding of the stringy

Kähler moduli space

i :MKah(X) → Aut(Db(X))\Stab+(Db(X))/C.

Moreover, the pullback of our Weil–Petersson metric is a non-degenerate Kähler metric onMKah(X).

It should be identified with the Weil–Petersson metric on the complex moduli spaceMcpx(Y) of a

mirror manifold of Y.

4

Chapter 3: Systolic inequality on K3 surfaces

Systolic geometry studies the least length of a non-contractible loop sys(M, g) in a

Riemannian manifold M. Loewner’s torus systolic inequality states that sys(T2, g)2 ≤2√3vol(T2, g) holds for any metric on the two-torus. We propose the following question that

naturally generalizes Loewner’s torus systolic inequality from the perspective of Calabi–Yau

geometry.

Question 0.0.9 (= Question 3.1.2). Let Y be a Calabi–Yau manifold, and let ω be a symplectic

form on Y. Does there exist a constant C > 0 such that

minL:sLag

∣∣∣ ∫L

Ω∣∣∣2 ≤ C ·

∣∣∣ ∫Y

Ω ∧Ω∣∣∣

holds for any holomorphic top form Ω on Y? Here “sLag" denotes the special Lagrangian submanifolds

in Y with respect to ω and Ω.

Motivated by the connection between flat surfaces and stability conditions, as well as the

conjectural description of Bridgeland stability conditions on Fukaya category by Bridgeland

and Joyce [14, 16, 55], we define the categorical analogue of minL:sLag

∣∣∣ ∫L Ω∣∣∣, which can also

be regarded as the categorical analogue of systole (see Table 3.1).

Definition 0.0.10 (= Definition 3.1.3). Let D be a triangulated category, and σ be a Bridgeland

stability condition on D. Its systole is defined to be

sys(σ) := min|Zσ(E)| : E is σ−semistable.

Using the idea in Chapter 2, we also define the categorical analogue of holomorphic

volume∣∣∣ ∫Y Ω ∧Ω

∣∣∣ of a Calabi–Yau manifold Y.

Definition 0.0.11 (= Definition 3.2.7). Let Ei be a basis of the numerical Grothendieck group

N (D) and let σ = (Z ,P) be a Bridgeland stability condition on D. Its volume is defined to be

vol(σ) :=∣∣∣∑

i,jχi,jZ(Ei)Z(Ej)

∣∣∣,where (χi,j) = (χ(Ei, Ej))

−1 is the inverse matrix of the Euler pairings.

5

Then one can ask the following question, which should give the same answer as Question

0.0.9 assuming mirror symmetry.

Question 0.0.12. Let X be a Calabi–Yau manifold and D = DbCoh(X) be its derived category of

coherent sheaves. Does there exist a constant C > 0 such that

sys(σ)2 ≤ C · vol(σ)

holds for any σ ∈ Stab∗(D)? Here Stab∗(D) is a subset of Stab(D) whose double quotient by

Aut(D) and C gives the stringy Kähler moduli spaceMKah(X) of X.

Question 0.0.9 and Question 0.0.12 are mirror to each other in the sense that

• in Question 0.0.9, the symplectic form ω is fixed, and the question is asking for the

supremum of the ratio sys2/vol among all Ω ∈ Mcpx(Y); while

• in Question 0.0.12, the complex structure Ω is fixed, and the question is asking for the

supremum of the ratio sys2/vol among all [σ] ∈ Aut(D)\Stab∗(D)/C ∼= MKah(X).

Note that the ratio sys(σ)2/vol(σ) is invariant under the Aut(D)-action and the free

C-action.

The main result in this chapter is the following theorem.

Theorem 0.0.13 (= Theorem 3.1.5). Let X be a K3 surface of Picard rank one, with Pic(X) = ZH

and H2 = 2n. Then

sys(σ)2 ≤ (n + 1)vol(σ)

holds for any σ ∈ Stab∗(D).

Chapter 4: Entropy of autoequivalences

The topological entropy htop( f ) of an automorphism f is an important dynamical invariant

that measures the complexity of the dynamical system formed by the iterations of f . A

fundamental theorem by Gromov–Yomdin [44, 45, 106] states that if X is a compact Kähler

6

manifold and f : X → X is a holomorphic surjective map, then

htop( f ) = log ρ( f ∗).

Here f ∗ denotes the induced linear map on H∗(X; C), and ρ( f ∗) is its spectral radius.

Let D be a triangulated category and let Φ : D → D be an endofunctor. The notion of

categorical entropy hcat(Φ) has been introduced by Dimitrov–Haiden–Katzarkov–Kontsevich

[29]. It was shown by Kikuta–Takahashi [63] that if D = DbCoh(X) and Φ = Ł f ∗ is induced

by an automorphism f on X, then hcat(Ł f ∗) = htop( f ). They made the following conjecture

which can be viewed as the categorical analogue of Gromov–Yomdin theorem.

Conjecture 0.0.14 (Kikuta–Takahashi [63]). Let X be a smooth projective variety over C and let

Φ : Db(X)→ Db(X) be an autoequivalence. Then

hcat(Φ) = log ρ([Φ]).

Here [Φ] denotes the induced linear map on the numerical Grothendieck group of Db(X).

The main result of this chapter is the discovery of the first counterexamples of Conjecture

0.0.14.

Theorem 0.0.15 (= Theorem 4.1.1 and Proposition 4.1.4). Let X be a Calabi–Yau hypersurface

in CPd+1 and d ≥ 4 be an even integer. Consider the autoequivalence Φ := TOX (−⊗O(−1))

on Db(X), where TOX is the spherical twist [89] with respect to the structure sheaf OX. Then

hcat(Φ) > 0 = log ρ([Φ]).

This gives a counterexample to Conjecture 0.0.14. In fact, hcat(Φ) is the unique positive real number

λ > 0 satisfying ∑k≥1χ(O(k))

ekλ = 1.

One can consider spherical twists as “hidden symmetries" on a Calabi–Yau manifold X:

they are not induced by automorphisms on X, but rather correspond to Dehn twists along

Lagrangian spheres on a mirror Calabi–Yau Y under mirror symmetry [89]. Theorem 0.0.15

shows that because of the existence of hidden symmetries on Calabi–Yau manifolds, the

7

autoequivalences on the derived category Db(X) contain much richer dynamical information

than the automorphisms on X.

On the other hand, it sill is interesting to characterize the autoequivalences that satisfy

Conjecture 0.0.14. Along this direction, we show that the Pd-twists defined by Huybrechts–

Thomas [52] satisfy Conjecture 0.0.14. Note that Pd-twists are mirror to Dehn twists along

Lagrangian complex projective spaces.

Theorem 0.0.16 (= Theorem 4.5.5). Let X be a smooth projective variety of dimension 2d over C,

and let PE be the Pd-twist [52] of a Pd-object E ∈ Db(X). Then

hcat(PE ) = 0 = log ρ([PE ]).

Hence Conjecture 0.0.14 holds for Pd-twists.

8

Chapter 1

Mirror of Atiyah flop1

1.1 Introduction

Flop is a fundamental operation in birational geometry. By the work of Kollár [64], any

birational transformation of compact threefolds with nef canonical classes and Q-factorial

terminal singularities can be decomposed into flops.

Atiyah flop is the most well-known among many different kinds of flops. It contracts a

(−1,−1) curve and resolves the resulting conifold singularity by a small blow-up, producing

a (−1,−1) curve in another direction, see Figure 1.1.

Figure 1.1: The Atiyah flop.

In mirror symmetry, complex and symplectic geometries are dual to each other. Flop is

an important operation in complex geometry. It is natural to ask whether there is a mirror

1Co-authored with Hansol Hong, Siu-Cheong Lau and Shing-Tung Yau. Reference: [35].

9

operation in symplectic geometry. In this chaper we focus on the mirror of Atiyah flop.

SYZ mirror symmetry of a conifold singularity is well-known by the works of [46,

20, 76, 2, 21, 56]. A conifold singularity is given by u1v1 = u2v2 in C4. There are two

different choices of anti-canonical divisors which turn out to be mirror to each other, namely

D1 = u2v2 = 1 and D2 = (u2 − 1)(v2 − 1) = 0. Consider the resolved conifold

OP1(−1)⊕OP1(−1), with the divisor D2 deleted. Its SYZ mirror is given by the deformed

conifold (u1, v1, u2, v2, z) ∈ C4 × C× : u1v1 = z + q, u2v2 = z + 1. Here q is the Kähler

parameter of the resolved conifold, namely q = e−A where A is the symplectic area of the

(−1,−1) curve in the resolved conifold. The deformed conifold contains a Lagrangian

sphere whose image in the z-coordinate projection is the interval [−1,−q] ⊂ C. The

Lagrangian sphere is mirror to the holomorphic sphere in the resolved conifold.

Now take the Atiyah flop. The Kähler moduli of the resolved conifold is the punctured

real line R−0, consisting of two Kähler cones R+ and R− of the resolved conifold and its

flop respectively. A serves as the standard coordinate and flop takes A ∈ R+ to −A ∈ R−.

Thus the Atiyah flop amounts to switching A to −A, or equivalently q to q−1. As a result,

the SYZ mirror changes from u1v1 = z + q, u2v2 = z + 1 to u1v1 = z + q−1, u2v2 = z + 1.

However the above two manifolds are symplectomorphic to each other, and hence they

are just equivalent from the viewpoint of symplectic geometry. Unlike Atiyah flop in

complex geometry, the mirror operation does not produce a new symplectic manifold. It is

not very surprising since symplectic geometry is much softer than complex geometry.

In contrast to complex geometry, the mirror flop is just a symplectomorphism rather

than a new symplectic manifold. First observe that this symplectomorphism is non-trivial

(Section 1.4.1).

Proposition 1.1.1. Given a symplectic threefold (X, ω) and a Lagrangian three-sphere S ⊂ X, we

have another symplectic threefold (X†, ω†) with a corresponding Lagrangian three-sphere S† ⊂

X†, together with a symplectomorphism f (X,S) : (X, ω) → (X†, ω†). It has the property that

f (X†,S†) f (X,S) = τ−1S , where τS is the Dehn twist along the Lagrangian sphere S.

We shall regard X and X† as the same symplectic manifold using the above symplecto-

10

morphism f (X,S).

We need to endow a symplectic threefold with additional geometric structures in order

to make it more rigid, so that the effect of the mirror flop can be seen. In the above local

case, u1v1 = z + q, u2v2 = z + 1 and u1v1 = z + q−1, u2v2 = z + 1 simply have different

complex structures. However in general requiring the existence of a complex structure on a

symplectic manifold would be too restrictive. Friedman [37] and Tian [97] showed that there

are topological obstructions to complex smoothing of conifold points; Smith-Thomas-Yau

[92] found the mirror statement for topological obstructions to Kähler resolution of conifold

points.

In this chapter, we consider two kinds of geometric structures, namely Lagrangian

fibrations, and Bridgeland stability conditions on the derived Fukaya category. First consider

a symplectic threefold X equipped with a Lagrangian fibration π : X → B. Let S ⊂ X be a

Lagrangian sphere. We assume that π around S is given by a local model of Lagrangian

fibration on the deformed conifold, where S is taken as the vanishing sphere under a

conifold degeneration, see Definition 1.4.3. We call such a fibration to be conifold-like

around S. Then we make sense of the mirror flop by doing a local surgery around S and

obtain another Lagrangian fibration π† : X → B. (X and X† have been identified by the

above symplectomorphism ρX,S.)

Theorem 1.1.2. Given a symplectic threefold (X, ω) with a Lagrangian fibration π : X → B which

is conifold-like around a Lagrangian three-sphere S ⊂ X, there exists another Lagrangian fibration

π† : X → B with the following properties.

1. π† is also conifold-like around S.

2. The images of S under π and π† are the same, denoted by S. They are one-dimensional affine

submanifolds in B away from discriminant locus.

3. π† = π outside a tubular neighborhood of S. In particular the affine structures on B induced

from π and π† are identical away from a neighborhood of S.

4. The induced orientations on S from π and π† are opposite to each other.

11

We call the change from π to π† to be the A-flop of a Lagrangian fibration along S.

As a compact example, consider the Schoen’s Calabi–Yau, which admits a conifold-like

Lagrangian fibration around certain Lagrangian spheres by the work of Gross [47] and

Castaño-Bernard and Matessi [76]. Then we can apply the A-flop to obtain other Lagrangian

fibrations.

More generally we can consider the effect of A-flop along S on Lagrangian submanifolds

other than Lagrangian torus fibers. Given a Lagrangian submanifold L ⊂ X which has T2-

symmetry around S (see Definition 1.4.7), we can construct another Lagrangian submanifold

L† (which also has T2-symmetry around S) which we call to be the A-flop of L, with the

property that (L†)† equals to the inverse Dehn twist of L along S.

Then we can take A-flop of special Lagrangian submanifolds with respect to a certain

holomorphic volume form (if it exists). Formally we start with a Bridgeland stability

condition (Z,P) [14] on the derived Fukaya category, where Z is a homomorphism of the K

group to C, and P is a collection of objects in the derived Fukaya category which are said to

be stable. A stability condition (Z,P) is said to be geometric if there exists a holomorphic

volume form Ω such that Z is given by the period∫·Ω and P is a collection of graded

special Lagrangians with respect to Ω. A-flop should be understood as a change of stability

conditions (Z,P) 7→ (Z†,P†).

In this chapter we realize the above for the local deformed conifold in Section 1.6. We

obtain the following theorem in Section 1.6.6.

Theorem 1.1.3. Let X be the deformed conifold u1v1 = z + q, u2v2 = z + 1, z 6= 0 (where

q 6= 1). Equip X with the holomorphic volume form Ω = dz∧ du1 ∧ du2. There exists a collection P

of graded special Lagrangians which defines a geometric stability condition (Z,P) on X. Moreover

the flop (Z†,P†) also defines a geometric stability condition with respect to ( f (X,S))∗ΩX† where

f (X,S) : X → X† = u1v1 = z + 1, u2v2 = z + 1/q : z 6= 0 is the symplectomorphism in

Proposition 1.1.1 (and ΩX† = dz ∧ du1 ∧ du2 on X†).

Stability conditions for the derived Fukaya category were constructed for the An case by

Thomas [94], for certain local Calabi–Yau threefolds associated to quadratic differentials by

12

Bridgeland–Smith [17, 91], and for punctured Riemann surfaces with quadratic differentials

by Haiden–Katzarkov–Kontsevich [48]. In this chapter we construct stability conditions

on the derived Fukaya category of the deformed conifold by applying the mirror functor

construction in [24, 23]; in the mirror side we use the results of Nagao–Nakajima [74] about

stability conditions on the noncommutative resolved conifold (see Theorem 1.6.9).

Theorem 1.1.4 (see Theorem 1.6.4). The mirror construction in [23] applied to the deformed

conifold X produces the noncommutative resolved conifold A given by Equation (1.15). In particular,

there is a natural equivalence of triangulated categories

Ψ : DbF → DbnilmodA (1.1)

where F is a subcategory of Fuk(X) generated by Lagrangians spheres, and DbnilmodA is a

subcategory of DbmodA consisting of modules with nilpotent cohomology.

The relation between the mirror construction in [23] and the SYZ construction is sum-

marized in Figure 1.2. The SYZ construction uses Lagrangian torus fibration coming from

degeneration to the large complex structure limit. The noncommutative mirror construction

in [23] uses Lagrangian vanishing spheres coming from degeneration to the conifold point.

LCSL of the deformed conifold

Another LCSL

conifold

B-side moduli of the local conifold

LVL of the resolved conifold

LVL of its op

nc resolution of the conifold

A-side moduli of the local conifold

SYZ

nc mirror construction

SYZ

Figure 1.2: The local conifold is self-mirror. More precisely for the local conifold, a resolution and its flop areequivalent, so the upper hemisphere should be identified with the lower hemisphere.

We shall prove that stable modules in DbnilmodA with respect to a certain stability

condition can be obtained as transformations of special Lagrangians under (1.1); as a result

the corresponding stability condition on DbF is geometric.

13

1.2 Review on flops and Bridgeland stability conditions

In this section, we recall the results by Toda which relate flops with wall-crossings in the

space of Bridgeland stability conditions on certain triangulated categories. For more details

and proofs, see [99].

1.2.1 Bridgeland stability conditions and crepant small resolutions

Let f : Y → Y be a crepant small resolution in dimension three and C the exceptional

locus, which is a tree of rational curves C = C1 ∪ · · · ∪ CN .

Define the triangulated subcategory DY/Y ⊂ Db(Y) to be

DY/Y := E ∈ Db(Y) | Supp(E) ⊂ C. (1.2)

Let pPer(Y/Y) ⊂ Db(Y) (p = 0,−1) be the abelian categories of perverse coherent

sheaves introduced by Bridgeland [13], and

pPer(DY/Y) :=p Per(Y/Y) ∩DY/Y.

Proposition 1.2.1 ([28]). The abelian categories 0Per(DY/Y) and −1Per(DY/Y) are the hearts of

certain bounded t-structures on DY/Y, and are finite-length abelian categories. The simple objects in

0Per(DY/Y) and −1Per(DY/Y) are ωC[1],OC1(−1), . . . ,OCN (−1) and OC,OC1(−1)[1], . . . ,

OCN (−1)[1] respectively.

Theorem 1.2.2 ([13][22]). Let g : Y† → Y be the flop of f , and φ : Y 99K Y† be the canonical

birational map. Then the Fourier-Mukai functor with the kernel OY×YY† ∈ Db(Y × Y†) is an

equivalence

ΦOY×YY†

Y→Y† : Db(Y)∼=−→ Db(Y†).

This equivalence restricts to an equivalenceDY/Y∼=−→ DY†/Y and takes 0Per(Y/Y) to −1Per(Y†/Y).

Such an equivalence is called standard in [99].

Let FM(Y) be the set of pairs (W, Φ), where W → Y is a crepant small resolution, and

Φ : Db(W)→ Db(Y) can be factorized into standard equivalences and the auto-equivalences

14

given by tensoring line bundles. For each (W, Φ) ∈ FM(Y), there is an associated open

subset

U(W, Φ) ⊂ Stabn(Y/Y)

of the space of normalized Bridgeland stability conditions on DY/Y. A Bridgeland stability

condition on DY/Y is called normalized if the central charge Z([Ox]) of the skyscraper sheaf

at each x ∈ C is −1.

Assume in addition that there is a hyperplane section in Y containing the singular point

such that its pullback in Y is a smooth surface, Toda proved the following theorem.

Theorem 1.2.3 ([99]). Let Stabn(Y/Y) be the connected component of Stabn(Y/Y) containing the

standard region U(

Y, Φ = idDb(Y)

). Define the following union of chambers

M :=⋃

(W,Φ)∈FM(Y)

U(W, Φ).

Then M ⊂ Stabn(Y/Y), and any two chambers are either disjoint or equal. Moreover, M =

Stabn(Y/Y).

In other words, we can obtain the whole connected component Stabn(Y/Y) from the

standard region U(Y, id) by sequence of flops and tensoring line bundles.

1.2.2 The conifold

Let Y = Spec C[[x, y, z, w]]/(xy− zw) and f : Y → Y be the blowing up at the ideal

(x, z). As computed in [99],

Stabn(Y/Y)/Aut0(DY/Y)∼= P1 − 3 points .

Let Y† → Y be the blowing up at the other ideal (x, w). Then the three removed points

correspond to the large volume limit points of Y and Y†, and the conifold point.

More precisely, P1− 3 points is obtained by gluing the upper and lower half complex

planes H, H†, and the real line with the origin removed. The hearts of the Bridgeland

stability conditions in H and H† are given by CohY/Y and CohY†/Y respectively. The

15

heart of the Bridgeland stability conditions on the real line is given by the perverse heart

0Per(DY/Y)∼=−1 Per(DY†/Y).

Let C, C† be the exceptional curves of Y → Y, Y† → Y respectively. Then the equivalence

DY/Y −→ DY†/Y satisfies

1. Φ(OC(−1)) = OC†(−1)[1].

2. Φ(OC(−2)[1]) = OC† .

3. For x ∈ C, the cohomology of E := Φ(Ox) ∈ DY†/Y vanish except for H0(E) = OC†

and H−1(E) = OC†(−1).

One can observe the following wall-crossing phenomenon: the skyscraper sheaves

Ox ∈ DY/Y are stable objects with respect to the stability conditions on the upper half plane

H, but are unstable in H†. In fact, its image under Φ is a two term complex E that fits into

the following exact triangle:

OC†(−1)[1]→ E→ OC†[1]→ (1.3)

Note that the usual skyscraper sheaf at a point in C† can be obtained by switching the first

and the third terms in (1.3).

Remark 1.2.4. It is well-known that if C is a (−1,−1)-curve, then the ‘flop-flop’ functor is the same

as the inverse of the spherical twist by OC(−1), i.e. ΦOY×YY†

Y†→YΦ

OY×YY†

Y→Y† = T−1OC(−1). Proposition

1.1.1 is the mirror statement of this fact.

1.3 Review on the SYZ mirror of the conifold

SYZ mirror construction for toric Calabi–Yau manifolds was carried out in [20] using the

wall-crossing techniques of [6]. The reverse direction, namely SYZ construction for blow-up

of V ×C along a hypersurface in a toric variety V was carried out by [2]. In this section we

recall the construction for the conifold Y = (u1, v1, u2, v2) ∈ C4 : u1v1 = u2v2 as a special

16

case in [20, 2]. The statement is that Y− u2v2 = 1 is mirror to Y− (u2 = 1 ∪ v2 = 1).

The study motivates the definition of A-flop for Lagrangian fibrations in the next section.

The resolved conifold Y = OP1(−1)⊕OP1(−1) is obtained from a small blowing-up of

the conifold point (u1, v1, u2, v2) = 0. It is a toric manifold equipped with a toric Kähler form.

We have the T2-action on Y given by (λ1, λ2) · (u1, v1, u2, v2) = (λ1u1, λ−11 v1, λ2u2, λ−1

2 v2),

and we denote the corresponding moment map by (µ1, µ2) : Y → R2. Then from the works

of Ruan [83], Gross [46] and Goldstein [42], there is a Lagrangian fibration

(µ1, µ2, |zw− 1|) : Y → R2 ×R≥0.

It serves as one of the local models of Lagrangian fibrations which were used by Castaño-

Bernard and Matessi [75, 76] to build up global fibrations from a tropical base manifold.

Figure 1.3: The base and discriminant loci of Lagrangian fibrations in conifold transition.

The discriminant locus of this fibration is contained in the hyperplane

(x1, x2, x3) ∈ R2 ×R≥0 : x3 = 1,

see the top left of Figure 1.3. This hyperplane is known as the wall for open Gromov–

Witten invariants of torus fibers as it contains images of holomorphic discs of Maslov index

zero. By studying wall-crossing of holomorphic discs emanated from infinity divisors (of a

compactification of Y), [20] constructed the SYZ mirror of Y− zw = 1.

Theorem 1.3.1 (A special case in [20] and [2]). The SYZ mirror of Y− u2v2 = 1 is

(u1, v1, u2, v2) : u1, v1 ∈ C, u2, v2 ∈ C× : u1v1 = 1 + u2 + v2 + qu2v2

17

where q = exp−(complexified symplectic area of the zero section P1 of Y

).

Take the change of coordinates u2 = q1/2u2 + 1/q1/2, v2 = q1/2v2 + 1/q1/2. (Here we

have fixed a square root of q.) Then the equation becomes u1v1 = u2v2 + 1− 1/q and the

divisors are u2 = 1/q1/2 and v2 = 1/q1/2. Further rescaling (u1, v1, u2, v2) by q1/2, the SYZ

mirror is the deformed conifold

Y = (u1, v1, u2, v2) ∈ C4 : u1v1 = u2v2 + (q− 1)

with the divisor (u2 − 1)(v2 − 1) = 0 deleted. To conclude, we have the mirror pair

Y− u2v2 = 1 and Y− (u2 − 1)(v2 − 1) = 0.

Taking the Atiyah flop of the (−1,−1) curve in Y amounts to switching q to 1/q. As a

result, the mirror of Y− zw = 1 changes from

u1v1 = u2v2 + (q− 1) − (u2 − 1)(v2 − 1) = 0

to

u1v1 = u2v2 + (1/q− 1) − (u2 − 1)(v2 − 1) = 0

under flop on Y. However changing equation just results in a symplectomorphism. Thus

unlike the flop of a (−1,−1) curve, the mirror flop (of a Lagrangian vanishing sphere

in conifold degeneration) does ‘nothing’ to the symplectic manifold. We need additional

geometric structures to detect the mirror flop. For this local model it is obvious that they

can be distinguished by complex structures. In general we would like to consider geometric

structures in the symplectic category. This will be further studied in the next section.

We can also consider a different relative Calabi–Yau so that Lagrangian spheres can be

seen more easily. First rescale (u1, v1, u2, v2) so that Y is given as

u1v1 − u2v2 = q1/2 − q−1/2.

Rewrite Y as a double conic fibration,

Y = (u1, v1, u2, v2, z) ∈ C5 : u1v1 = z + q1/2, u2v2 = z + q−1/2.

18

It is equipped with the standard symplectic form from C5. If we flop Y, the mirror Y

becomes u2v2 = z + q−1/2; u1v1 = z + q1/2. We take the complement Y− z = c where

c ∈ C− −q1/2,−q−1/2,

We have the Lagrangian fibration

(x1, x2, x3) = (|u1|2 − |v1|2, |u2|2 − |v2|2, |z− c|) : Y → R2 ×R≥0

where the boundary divisor is exactly z = c. The discriminant loci are x1 = 0, x3 =

|q1/2 + c| and x2 = 0, x3 = |q−1/2 + c| contained in the walls x3 = |q1/2 + c| and

x3 = |q−1/2 + c| respectively, see the top right of Figure 1.3. By [2, Theorem 11.1] (or SYZ

in [68] by Minkowski decompositions), the resulting SYZ mirror is the following.

Theorem 1.3.2 (A special case in [2] and [68]). The SYZ mirror of Y− z = c is Y− (u2 =

1 ∪ v2 = 1).

Denote a = −q−1/2 and b = −q1/2, and without loss of generality assume that a, b are

real, c = 0 and a < b < 0. Consider the Fukaya category of Y− z = 0 generated by the

two Lagrangian spheres S1 and S2, where

S0 =z = −t, |u1| = |v1|, |u2| = |v2| : a ≤ t ≤ b,

S1 =z = exp(tζ1 + (1− t)ζ0) for t ∈ [0, 1], |u1| = |v1|, |u2| = |v2|

where ζ0 = log |a| − πi and ζ1 = log |b| + πi . S0 and S1 are oriented by dt ∧ dθ1 ∧ dθ2

where θ1, θ2 are the arguments of u1, u2 respectively. They are special Lagrangians and in

particular graded by a suitable holomorphic volume form. (We shall go back to this point in

more detail in Section 1.5.) Figure 1.4 shows S0 in the picture of double conic fibration.

Chan–Pomerleano–Ueda [21] proved homological mirror symmetry for the mirror pair

(Y− z = 0, Y− (u2 = 1 ∪ v2 = 1) making use of the SYZ transformation. The result

is the following.

Theorem 1.3.3 (Theorem 1.2 and 1.3 of [21]). There is an equivalence between the derived

wrapped Fukaya category of Y0 := Y − z = 0 and the derived category of coherent sheaves of

Y0 := Y− (u2 = 1 ∪ v2 = 1).

19

Figure 1.4: Lagrangian S3 seen from the double conic fibration.

Remark 1.3.4. The spheres S0 and S1 here were denoted as S1 and S0 in [21] respectively.

In Section 1.5 and 1.6, Y0 will be denoted as Xt=0 which appears as a member in a family

of symplectic manifolds Xt.

Restricting to the Fukaya subcategory consisting of S0, S1, we have the equivalence

between Db〈S0, S1〉 and DY/Y (1.2). We will revisit this equivalence in Section 1.5 (see

Theorem 1.5.3 for more details on the equivalence). Then we will compare the flop on B-side

and the corresponding operation on A-side (to be constructed below) using this.

On the other hand, we can take the approach of [23] to construct the noncommutative

mirror of Y0. From homological mirror symmetry between Y0 and its noncommutative

mirror, we obtain stability conditions on the derived Fukaya category generated by S0 and

S1 in Section 1.6. We will show that stable objects are special Lagrangian submanifolds.

1.4 A-flop in symplectic geometry

1.4.1 Mirror of Atiyah flop as a symplectomorphism

Let (X, ω) be a symplectic threefold and S a Lagrangian sphere of X. By Weinstein

neighborhood theorem, a neighborhood of S ⊂ X can always be identified symplectomor-

20

phically with a neighborhood of S ⊂ T∗S, which can be identified with (u1, v1, u2, v2) ∈

C4 : u1v1 − u2v2 = ε = (u1, v1, u2, v2, z) ∈ C5 : u1v1 = z + ε, u2v2 = z for some ε > 0,

where ω is given by the restriction of the standard symplectic form on C4. This identification

is adapted to conifold degeneration at the limit ε→ 0.

In other words, we take a conifold-like chart in the following sense.

Definition 1.4.1. A conifold-like chart around S is (U, ι), where U is an open neighborhood of S

and ι : U → C× ×C4 is a symplectic embedding (where C× ×C4 is equipped with the standard

symplectic form) such that the following holds.

1. The image of U under the embedding is given by u1v1 = z− a,

u2v2 = z− b(1.4)

for some real numbers a < b, where∣∣∣z− a+b

2

∣∣∣ < R for some fixed R > b−a2 , and

∣∣|u1|2 − |v1|2∣∣ <

L,∣∣|u2|2 − |v2|2

∣∣ < L for some fixed L > 0. Here z is the coordinate of C× and u1, v1, u2, v2

are the coordinates of C4. We will also denote the image by U for simplicity.

2. The Lagrangian sphere ι(S) is given by |u1| = |v1|, |u2| = |v2|, z ∈ [a, b].

We will simply identify U with its image under ι. Let

V =

u1v1 = z− a, u2v2 = z− b,

∣∣∣∣z− a + b2

∣∣∣∣ ≤ R− ε,∣∣|ui|2 − |vi|2

∣∣ ≤ L− ε for i = 1, 2⊂ U,

V ′ =

u1v1 = z− a, u2v2 = z− b,∣∣∣∣z− a + b

2

∣∣∣∣ < R− 2ε,∣∣|ui|2 − |vi|2

∣∣ < L− 2ε for i = 1, 2⊂ V

for ε > 0 sufficiently small. We have a diffeomorphism from U −V to the corresponding

open subset of

U† :=

u1v1 = z− b, u2v2 = z− a,∣∣∣∣z− a + b

2

∣∣∣∣ < R, for i = 1, 2

defined by z 7→ z, (u1, v1) 7→(

z−bz−a

)1/2(u1, v1), (u2, v2) 7→

( z−az−b

)1/2(u2, v2). (We choose a

branch of the square root. It is well-defined since a, b /∈ U −V.) By Moser argument, we

21

can cook up a symplectomorphism isotopic to this diffeomorphism. Thus we have fixed a

symplectomorphism ρ from U −V to an open subset of U†.

In analogous to a flop along a (−1,−1) curve in complex geometry, we define an-

other symplectic threefold (X†, ω†) by gluing (X − V, ω) with a suitable open subset

of U† by the above symplectomorphism ρ on U − V. By construction (an open sub-

set of) U† is a conifold chart of X† around the Lagrangian sphere S† ⊂ U† defined by

|u1| = |v1|, |u2| = |v2|, z ∈ [a, b].

However, unlike the flop along a (−1,−1) holomorphic sphere, (X†, ω†) is just sym-

plectomorphic to the original (X, ω), since the gluing map ρ can be extended to U → U†.

Let

ψ± : (u1, v1, u2, v2, z) ∈ C5 : u1v1 = z− a, u2v2 = z− b → u1v1 = z− b, u2v2 = z− a

be defined by (u1, v1, u2, v2, z) 7→ (±i u1,±i v1,±i u2,±i v2,−z + a + b) respectively. It

commutes with the T2 action

(λ1, λ2) · (u1, v1, u2, v2, z) = (λ1u1, λ−11 v1, λ2u2, λ−1

2 v2, z)

and hence descends to the symplectic reduction, which is simply rotating the z-plane by π

around (a + b)/2. Let ψ be the restriction of ψ+ to V ′. Then we have a symplectomorphism

U → U† by interpolating between the gluing map ρ and ψ in the region R− 2ε <∣∣∣z− a+b

2

∣∣∣ <R− ε, L− 2ε <

∣∣|ui|2 − |vi|2∣∣ < L− ε. Namely we take a diffeomorphism which equals to

ψ on V ′, and is given by

z 7→eπi f (|z−(a+b)/2|)(z− (a + b)/2) + (a + b)/2

(u1, v1) 7→(

eπi f (|z−(a+b)/2|)(z− (a + b)/2) + (a− b)/2z− a

)1/2

(u1, v1)

(u2, v2) 7→(

eπi f (|z−(a+b)/2|)(z− (a + b)/2) + (b− a)/2z− b

)1/2

(u2, v2)

on U−V. Here f (r) is a decreasing function valued in [0, 1] which equals to 1 for r < R− 2ε

and equals to 0 for r > R− ε. The square root z1/2 is taken for the branch 0 < arg(z) ≤ π.

22

By Moser argument we have a symplectomorphism isotopic to this, and ρ is the restriction

to U −V.

In conclusion, given a symplectic manifold (X, ω) and a conifold-like chart around a

Lagrangian sphere S, we have a symplectomorphism f (X,S) : (X, ω)→ (X†, ω†) by a surgery

in analogous to flop in complex geometry. The operation does not produce a new symplectic

manifold because symplectic geometry is too soft.

If we do the operation twice, we obtain X†† which is canonically identified with X as fol-

lows. X†† is glued from X−V and U = U†† by ρ† ρ. The composition of z 7→ z, (u1, v1) 7→(z−bz−a

)1/2(u1, v1), (u2, v2) 7→

( z−az−b

)1/2(u2, v2) and z 7→ z, (u1, v1) 7→

( z−az−b

)1/2(u1, v1),

(u2, v2) 7→(

z−bz−a

)1/2(u2, v2) is simply identity. Hence the gluing ρ† ρ = Id and X†† = X.

Below we see that doing the above operation twice produces the Dehn twist along the

Lagrangian sphere S, which induces a non-trivial automorphism on the Fukaya category.

Proposition 1.4.2 (same as Proposition 1.1.1). f (X†,S†) f (X,S) : X → X†† = X equals to the

inverse of the Dehn twist of X along S.

Proof. f (X†,S†) f (X,S) : X → X†† is given as follows. Write X = X†† = (X −V) ∪Id U. The

map is identity on X−V. In V ′ ⊂ U it is given by ψ2 which maps ui 7→ −ui, vi 7→ −vi, z 7→ z,

and in particular is the antipodal map on the three-sphere

u1v1 = z− a, u2v2 = z− b, z ∈ [b, a], |ui| = |vi| for i = 1, 2 ⊂ V ′.

On U −V ′ it is isotopic to

z 7→e2πi f (|z−(a+b)/2|)(z− (a + b)/2) + (a + b)/2

(u1, v1) 7→(

e2πi f (|z−(a+b)/2|)(z− (a + b)/2) + (b− a)/2z− a

)1/2

(u1, v1)

(u2, v2) 7→(

e2πi f (|z−(a+b)/2|)(z− (a + b)/2) + (a− b)/2z− b

)1/2

(u2, v2),

where f (r) is a decreasing function valued in [0, 1] which equals to 1 for r < R− 2ε and

equals to 0 for r > R− ε. The square root z1/2 is taken for the branch where 0 < arg(z) ≤ 2π.

23

Thus we see that it is the inverse of the Dehn twist.

1.4.2 Lagrangian fibrations

We see from the last section that the mirror of the Atiyah flop surgery does not produce

a new symplectic manifold unfortunately. We need additional geometric structures in order

to distinguish X† from X. In this section we consider Lagrangian fibrations. Conceptually

it can be understood as a ‘real polarization’, playing the role of the complex polarization

(namely the complex structure) for flop of a (−1,−1) curve.

From now on we identify X and X† as the same symplectic manifold using the symplec-

tomorphism f (X,S).

Let π : X → B be a Lagrangian torus fibration. We consider a conifold degeneration of

X with a vanishing sphere S, such that the Lagrangian fibration around S is like the one on

the deformed conifold [46, 42].

Definition 1.4.3. Assume the notations in Section 1.4.1. A Lagrangian fibration π is said to be of

conifold-like if we have the commutative diagram

U ι(U)

f (U) I × (−L, L)× (−L, L)?

π

-ι

?(|z−c|, 1

2 (|u1|2−|v1|2), 12 (|u2|2−|v2|2))

-∼=

where∣∣∣c− a+b

2

∣∣∣ > R and |c− a| 6= |c− b|, and I is a certain open interval.

Theorem 1.4.4 (Theorem 1.1.2 in the Introduction). Given a symplectic threefold (X, ω) with a

Lagrangian fibration π : X → B which is conifold-like around a Lagrangian three-sphere S ⊂ X,

there exists a Lagrangian fibration π† : X → B with the following properties.

1. π† is also conifold-like around S.

2. The images of S under π and π† are the same, denoted by S. It is a one-dimensional affine

submanifold in B away from discriminant locus.

24

3. π† = π outside a neighborhood V ⊃ S and V ⊂ U. In particular the affine structures on B

induced from π and π† are identical away from a neighborhood of S.

4. From above, there is a canonical correspondence between orientations of regular fibers of π and

that of π†. Fix an orientation of torus fibers of π over the image of U, and an orientation of a

regular fiber of π|S (which is topologically T2). Then the induced orientations on S through π

and π† are opposite to each other.

Proof. π† is constructed from the symplectomorphism f : X → X† given in the last subsec-

tion. Namely we glue the Lagrangian fibration of X−V with the Lagrangian fibration of U†

by ρ. This gives a Lagrangian fibration on X†, and hence on X by the symplectomorphism

f . It is constructed directly as follows.

For each fixed u1, u2, v1, v2, take the diffeomorphism φu1,u2,v1,v2 on z ∈ C : |z − (a +

b)/2| < R defined by

φu1,u2,v1,v2(z) = eπi f

((|z−(a+b)/2|

R

)2+

(|u1 |

2−|v1 |2

L

)2

+

(|u2 |2−|v2 |2

L

)2)(z− (a + b)/2) + (a + b)/2 (1.5)

where f (r) is a decreasing function valued in [0, 1] which equals to 1 for r < 1− 2ε and

equals to 0 for r > 1− ε. Thus φu1,u2,v1,v2 is identity on(u1, v1, u2, v2, z) ∈ U :

(|z− (a + b)/2|

R

)2

+

(|u1|2 − |v1|2

L

)2

+

(|u2|2 − |v2|2

L

)2

> 1− ε

.

Define a fibration U → I × (−L, L)× (−L, L) by(|φu1,u2,v1,v2 (z)− c| , 1

2(|u1|2 − |v1|2),

12(|u2|2 − |v2|2)

).

For |u1| = |v1|, |u2| = |v2|, the resulting level curves of |φ(0, 0, z)− c| are depicted in Figure

1.5. Since the fibration is T2-equivariant and any curve on the plane is Lagrangian, it

is a Lagrangian fibration by symplectic reduction. Moreover it agrees with the original

Lagrangian fibration π on U − V. Hence we can glue this with the original Lagrangian

fibration on X−V, and obtain another Lagrangian fibration π† : X → B.

By definition π = π† away from V. In the neighborhood defined by(|z−(a+b)/2|

R

)2+

25

Figure 1.5: The symplectic reduction of Lagrangian torus fibers before and after the A-flop.

(|u1|2−|v1|2

L

)2+(|u2|2−|v2|2

L

)2< 1− 2ε, the fibration is simply(

|z− (a + b− c)| , 12(|u1|2 − |v1|2),

12(|u2|2 − |v2|2)

)which is also conifold-like around S. The image of S under either π and π† is the in-

terval [b − c, a − c] × 0 × 0. Since the fibration around S is compatible with the

symplectic reduction of the T2-action on (u1, v1, u2, v2), the second and third coordinates

(b2, b3) =( 1

2 (|u1|2 − |v1|2), 12 (|u2|2 − |v2|2)

)serve as the action coordinates of the base of

the Lagrangian fibration. Thus the image S ⊂ b2 = b3 = 0 is an affine submanifold.

Since π = π† outside V ⊂ U and every torus fiber of π and π† has non-empty

intersection in X − V, the orientations can be canonically identified. We have induced

orientations on the fibers of |z − c| and also fibers of |φu1,u2,v1,v2(z) − c|. The induced

orientation on S from π (or π† resp.) is such that ω(u, v) > 0 where u is a tangent vector

along the orientation of S and v (or v′ resp.) is a tangent vector along the orientation of a

fiber of |z− c| (or a fiber of |φu1,u2,v1,v2(z)− c| resp.). It follows that v = −v′ (up to scaling

by a positive number) and hence the two induced orientations on S are opposite to each

other.

To distinguish from the usual notion of flop in complex geometry, we call π† to be the

A-flop of the Lagrangian fibration π (where ‘A’ stands for the ’symplectic side’ in mirror

26

symmetry). It is the mirror operation of Atiyah flop.

In analogous to foliations, we identify two Lagrangian fibrations if they are related by

diffeomorphisms as follows.

Definition 1.4.5. Two Lagrangian fibrations π1, π2 : X → B are said to be equivalent if there exists

a symplectomorphism Φ : X∼=→ X and a diffeomorphism φ : B

∼=→ B such that φ π1 = π2 Φ.

The following easily follows from construction.

Proposition 1.4.6. If we make different choices of (U, ι) and the function f in the proof of Theorem

1.4.4, the resulting Lagrangian fibrations are equivalent.

1.4.3 A-flop on Lagrangian submanifolds

We can also consider the effect of A-flop on Lagrangian submanifolds other than torus

fibers. We restrict to the following kind of Lagrangian submanifolds.

Definition 1.4.7. Let S be a Lagrangian sphere in X. A Lagrangian submanifold L ⊂ X is said to

have T2-symmetry around S if there exists a conifold-like chart (U, (u1, v1, u2, v2, z)) around S such

that the image of L∩U under z is a curve and the fiber at each point is given by |ui|2− |vi|2 = ci(z)

for some real-valued function ci on the curve, i = 1, 2.

Given L with T2-symmetry with respect to a conifold-like chart (U, (u1, v1, u2, v2, z))

around S, we define L† as follows. Recall the diffeomorphism φu1,u2,v1,v2 on z ∈ C :

|z− (a + b)/2| < R in Equation (1.5). Write the image of L ∩U under z as a level curve

f (z) = 0 of a real-valued function f . Then L† is given as

(u1, v1, u2, v2, z) ∈ U : f (φu1,u2,v1,v2(z)) = 0, |ui|2 − |vi|2 = ci(z) for i = 1, 2

in U and equals to L outside U. Since the image of L† is a curve in the symplectic reduction

by T2, it is a Lagrangian submanifold with T2-symmetry. Note that L† and L can be

topologically different from each other.

By construction we have

27

Proposition 1.4.8. (L†)† equals to the inverse of the Dehn twist applied to L.

If we have a stability condition (Z,P) on the Fukaya category generated by Lagrangians

with T2-symmetry around S, then A-flop should give another stability condition (Z†,P†)

where Z†(L†) = Z(L). In the next two sections we will restrict to the deformed conifold

and carry out this construction explicitly.

1.4.4 Examples

Deformed conifold

Consider the deformed conifold X = (u1, v1, u2, v2, z) ∈ C4 ×C× : u1v1 = z− a, u2v2 =

z− b where a < b < 0. (We have taken away the divisor z = 0.) We can take the flop of the

Lagrangian fibration

π = (|u1|2 − |v1|2, |u2|2 − |v2|2, |z|)

which is special with respect to the holomorphic volume form Ω = d log z ∧ du1 ∧ du2. The

base of the fibration is R2 ×R>0. The discriminant locus of the fibration is 0 ×R×

|a| ∪R× 0 × |b|. After the flop, the discriminant locus becomes (0, t, |φ0,t,0,0(a)|) :

t ∈ R ∪ (t, 0, |φ0,t,0,0(b)|) : t ∈ R where φ is given in Equation (1.5). For t 1,

φ0,t,0,0(a) = b and φ0,t,0,0(b) = a; for t big enough, φ0,t,0,0(a) = a and φ0,t,0,0(b) = b. The

base and discriminant locus are shown in Figure 1.6. The new fibration π† is no longer

special with respect to Ω; however it is equivalent to the corresponding special Lagrangian

fibration on X† = (u1, v1, u2, v2, z) ∈ C4 ×C× : u1v1 = z− b, u2v2 = z− a with respect to

Ω† (defined by the same expression as Ω).

We have a family of complex manifolds defined by

u1v1 =z−(

a + b2

+b− a

2eπi (1+s)

),

u2v2 =z−(

a + b2

+b− a

2eπi s

)

for s ∈ [0, 1] joining X and X†. They are depicted in Figure 1.7. Each member has a special

28

Figure 1.6: The discriminant locus before and after the A-flop.

Lagrangian fibration defined by the same formula as π above. Before (or after) the moment

s = 12 , the Lagrangian fibrations are all equivalent. Approaching the moment s = 1

2 , two

singular Lagrangian fibers collide into one and the Lagrangian fibration changes. Thus

s = 12 is the ‘wall’. It can also been seen clearly from the base, see the top right of Figure 1.3.

At the moment s = 12 , the two discriminant loci (which are two lines in different directions)

collides.

Figure 1.7: A-flop shown in the double conic fibration picture.

The torus fibers of π and π† are different objects in the Fukaya category. Namely

consider a fiber T = |z| = k, |u1| = |v1|, |u2| = |v2| of π where |b| < k < |a| and the

corresponding fiber T† = |φ(z)| = k, |u1| = |v1|, |u2| = |v2| of π†. We shall see in

Section 1.5 that T, which is special Lagrangian with respect to Ω, is a surgery S1#S0 for

29

a morphism in Mor(S1, S0), while T†, which is special Lagrangian with respect to Ω†, is

a surgery S0#S1 for a morphism in Mor(S0, S1). S0, S1 are Lagrangian spheres defined by

S0 = z = −t, |u1| = |v1|, |u2| = |v2| : a ≤ t ≤ b and S1 = z = 1 + exp(tζ1 + (1−

t)ζ0) for t ∈ [0, 1], |u1| = |v1|, |u2| = |v2| where ζ0 = log |a| − i π and ζ1 = log |b|+ i π.

Deformed orbifolded conifold

For k ≥ l ≥ 1, the orbifolded conifold Ok,l is the quotient of the conifold u1v1 =

u2v2 ⊂ C4 by the abelian group Zk ×Zl , where the primitive roots of unity ζk ∈ Zk and

ζl ∈ Zl act by

(u1, v1, u2, v2) 7→ (ζku1, ζ−1k v1, u2, v2), and (x, y, z, w) 7→ (u1, v1, ζlu2, ζ−1

l v2).

In equations

Ok,l =

u1v1 = (z− 1)k, u2v2 = (z− 1)l⊂ C5.

It is a toric Gorenstein singularity whose fan is the cone over the rectangle [0, k]× [0, l] ⊂ R2.

For the purpose of constructing a Lagrangian torus fibration with only codimension-

two discriminant loci, we shall delete the divisor z = 0 ⊂ Ok,l and obtain X0 =(u1, u2, v1, v2, z) ∈ C4 ×C× : u1v1 = (z− 1)k, u2v2 = (z− 1)l .

We shall consider smoothings of X0, which correspond to the Minkowski decompositions

of the rectangle [0, k] × [0, l] into k copies of [0, 1] × 0 and l copies of 0 × [0, 1] [4].

Explicitly a smoothing is given by

X = (u1, u2, v1, v2, z) ∈ C4 ×C× | u1v1 = f (z), u2v2 = g(z)

where f (z) and g(z) are polynomials of degree k and l respectively, such that the roots ri

and sj of f (z) and g(z) respectively are pairwise-distinct and non-zero. For later purpose

we shall assume |ri|, |sj| are all pairwise distinct.

X admits a double conic fibration X → C× by projecting to the z-coordinate. There is also

a natural Hamiltonian T2-action on X given by (s, t) · (u1, v1, u2, v2, z) := (su1, s−1v1, tu2, t−1v2, z)

for (s, t) ∈ T2 ⊂ C2. The symplectic reduction of X by the T2-action is identified with C×,

30

the base of the double conic fibration. Using the construction of Goldstein [42] and Gross

[46], we have the Lagrangian fibration

π : X → B := R2 × (0, ∞)

π(u1, v1, u2, v2, z) =(

12(|u1|2 − |v1|2),

12(|u2|2 − |v2|2), |z|

).

The map to the first two coordinates is the moment map of the Hamiltonian T2-action. We

denote the coordinates of B by b = (b1, b2, b3). The discriminant locus is given by the disjoint

union of lines (k⋃

i=1

b1 = 0, b3 = |ri|)∪

l⋃j=1

b2 = 0, b3 = |sj|

⊂ B,

and the fibers are special Lagrangians in the same phase π/2 with respect to the volume

form Ω := du1 ∧ du2 ∧ d log z ([56, Proposition 3.17]).

Now let a = r1 and b = s1. Assume that |a| 6= |b|; zero and all other roots ri, sj lie outside

the disc |z− (a + b)/2|. Let S0 be the Lagrangian matching sphere corresponding to the

straight line segment joining a and b. Then the above Lagrangian fibration is conifold-like

around S. The flop of this is equivalent to the corresponding Lagrangian fibration on

X† = u1v1 = f †(z), u2v2 = g†(z), where f † and g† are polynomials with sets of roots

s1, r2, . . . , rk and r1, s2, . . . , sl respectively; and the Lagrangian fibration is

π†(u1, v1, u2, v2, z) =(

12(|u1|2 − |v1|2),

12(|u2|2 − |v2|2), |z|

): X† → B.

The Lagrangian fibration π† is no longer special with respect to Ω on X; however it is

(equivalent to) a special Lagrangian fibration with respect to Ω† = du1 ∧ du2 ∧ d log z on X†.

Schoen’s Calabi–Yau

Given a compact simple integral affine threefold B with singularities ∆, Castaño-Bernard

and Matessi [75] constructed a symplectic manifold X together with a Lagrangian fibration

X → B inducing the given affine structure. It is achieved by gluing local models of

Lagrangian fibrations around ∆ with the Lagrangian fibration over the affine manifold

31

B− ∆. In particular their construction can be applied to Schoen’s Calabi–Yau [76]. The

Lagrangian fibration is conifold-like, and so the mirror flop defined here can be applied.

Schoen’s Calabi–Yau is given by the fiber product of two elliptic fibrations on K3 surfaces

over the base P1. The affine base manifold (which is topologically S3) of Schoen’s Calabi–Yau

was found by Gross [47, Section 4]. Section 9.2 of [76] constructed a conifold degeneration

of the affine base forming nine conifold points simultaneously.

We quickly review their construction here. Consider the following polyhedral decompo-

sition of S3. Take six copies of triangular prisms

Conv(0, 0, 0), (0, 1, 0), (1, 0, 0), (0, 0, 1), (0, 1, 1), (1, 0, 1),

three of them are labeled as σj and three of them are labeled as τk for j, k ∈ Z3. Take nine

copies of cubes

Conv(0, 0, 0), (0, 1, 0), (1, 0, 0), (1, 1, 0), (0, 0, 1), (1, 0, 1), (0, 1, 1), (1, 1, 1)

and label them as ωjk. See Figure 1.8. The top triangular face of σj is glued to the bottom

triangular face of σj+1 (j ∈ Z3), and so topologically⋃3

j=1 σj forms a solid torus. Similarly

do the same thing for τk so that⋃3

k=1 τk forms another solid torus. For the nine cubes, glue

the top face of ωjk with the bottom face of ωj+1,k for j ∈ Z3, and glue the right face of

ωjk with the left face of ωj,k+1 for k ∈ Z3. This topologically forms a two-torus times an

interval. Finally glue the front face of ωjk with the j-th square face of τk, and glue the back

face of ωjk with the k-th square face of σj. Here the square faces of σj and τk are ordered

counterclockwisely. This forms S3 as gluing of two solid tori along their boundaries.

The fan structure at every vertex of the polyhedral decomposition is that of P2 ×P1.

Together with the standard affine structure of each polytope, this gives S3 an affine structure

with singularities. The discriminant locus is given by the dotted lines shown in Figure 1.8.

Note that each dotted line in a square face of a prism indeed has multiplicity three. Thus

the discriminant locus is a union of 24 circles counted with multiplicities. Moreover the

dotted lines in cubes form three horizontal and three vertical circles, intersecting with each

32

(0,0,0)

(0,0,1)

(1,0,0)

(1,0,1)

(0,1,0)

(0,1,1)glue

glue

glue

glue

The whole is a three-sphere topologically.

Figure 1.8: Polytopes in the polyhedral decomposition of the affine base of Schoen’s CY.

other at nine points. These are the nine conifold singularities (which are positive nodes).

By gluing local models of Lagrangian fibrations around discriminant locus with the

Lagrangian fibration from the affine structure away from discriminant locus, [76] produced a

symplectic manifold which is homeomorphic to the Schoen’s Calabi–Yau. Moreover by using

the results on symplectic resolution of Smith–Thomas–Yau [92] and complex smoothing of

Friedman [37] and Tian [97], they showed that the existence of certain tropical two-cycles

containing a set of conifold points ensure that the nodes can be simultaneously resolved

(and smoothened in the mirror side). In particular all the nine nodes in this example can be

resolved simultaneously.

In the smoothing the three horizontal and three vertical circles which form part of the

discriminant locus are moved apart so that they no longer intersect with each other. This

gives a symplectic manifold X together with a Lagrangian fibration. The corresponding

affine base coincides with the one in the previous work of Gross [47, Section 4].

The local model for each conifold point in this example is the Lagrangian fibration

(|u1|2 − |v1|2, |u2|2 − |v2|2, |z|) on (u1, v1, u2, v2, z) ∈ C4 × C× : u1v1 = z− a = u2v2 for

a < 0; the local model for its smoothing is the fibration defined by the same expression

on (u1, v1, u2, v2, z) ∈ C4 ×C× : u1v1 = z− a, u2v2 = z− b for a < b < 0. A Lagrangian

fibration corresponding to the simultaneous smoothing can be constructed by gluing these

33

local models. In particular X and the fibration are conifold-like around each of the van-

ishing spheres corresponding to the nine conifold points. Hence we can perform A-flop

around each of these spheres and obtain new Lagrangian fibrations. The operation can be

understood as link surgery in the base S3.

Note that we cannot always keep the circles Ai, Bj in constant levels in the A-flop. For

instance, suppose Ai and Bj are contained in the planes in levels ai, bj respectively with

a1 < a2 < a3 < b1 < b2 < b3. (These planes have normal vectors pointing to the right if

drawn in Figure 1.8.) Now we perform the A-flop along the vanishing sphere between

levels a1 and b1. The resulting fibration is equivalent to the one with these circles in levels

a2 < a3 < b1 < a′1 < b2 < b3 where a′1 is the new level of A1. At this stage all these circles

are still kept in constant levels. Now let’s do the A-flop along the vanishing sphere between

levels a2 and b3. Then the resulting fibration cannot have all these circles in constant levels:

if they were in constant levels, then a2 < b1 < a′1 < b3 < a2, a contradiction!

1.5 Derived Fukaya category of the deformed conifold

In Example 1.4.4, we consider a path of complex structures on the deformed conifold

(with a fixed symplectic form) given by the equations

Xs =

u1v1 = z−

(a + b

2+

b− a2

eπi (1+s))

, u2v2 = z−(

a + b2

+b− a

2eπi s

), z 6= 0

(1.6)

for s ∈ [0, 1]. (Xs=0 and Xs=1 were denoted as X and X† in 1.4.4, respectively.) This

deformation of complex structures parametrized by s is SYZ mirror to the flop operation

on the resolved conifold. In the last section we realized this operation as surgery of a

Lagrangian fibration.

In this section, we study the effect of deformation of complex structures (together with

holomorphic volume forms) on special Lagrangians. This would motivate us to consider

A-flop on stability conditions of the derived Fukaya category.

Recall from Section 1.3 that we have two Lagrangian spheres S0 and S1 in Xs=0. Moreover,

34

there is a sequence of Lagrangian spheres Sn : n ∈ Z in Xs=0 which corresponds to a

collection of non-trivial matching paths in the base of the double conic fibration Xs=0 → C×.

We depict these spheres in the universal cover of C×(3z) as shown in Figure 1.9.

Figure 1.9: Sequence of Lagrangian spheres in Xs=0.

Definition 1.5.1. F is defined to be the full subcategory of Fuk(Xs=0) generated by S0 and S1.

The main purpose of this section is to prove the following.

Theorem 1.5.2. Regular Lagrangian torus fibers of π which have non-empty intersection with S0,

as well as the Lagrangian spheres Si, are contained in F .

The theorem follows from Proposition 1.5.7 and 1.5.9 below.

The torus fibers and spheres Si are special Lagrangians with respect to the holomorphic

volume form

Ω := d log z ∧ du1 ∧ du2

on Xs=0. Here we used d log z instead of d log z to match the ordering of phases on both

sides of the mirror2. In particular, we measure the angle in clockwise direction for phases of

Si. The diagram in the right side of Figure 1.9 compares the phases of Si’s, where S0 has the

2In order to match the phase inequality in the mirror side, we can either impose the mirror functor to becontravariant, or use the complex structure induced by the conjugate volume form d log z ∧ du1 ∧ du2 like here.All Si are still special Lagrangians under this volume form, and we have the phase inequalities θ(Si) > θ(Sj) for0 < i < j or i < j < 1. This matches the ordering of the phases of stable objects in an exact triangle of the mirror

B-side convention. Namely for an exact triangle L1 → L1#L2 → L2[1]→ where Li are special Lagrangians, their

phases should satisfy θ(L1) ≤ θ(L1#L2) ≤ θ(L2).

35

biggest phase in our convention. In Section 1.6 we will see that taking these to be stable

objects defines a Bridgeland stability condition on the derived Fukaya category.

Moreover each Si corresponds to another Lagrangian sphere S†i in F , the flop of Si

constructed in Section 1.4.3. The Lagrangian torus fibers of π† and S†i are special with

respect to the pull-back holomorphic volume form from Xs=1, and they define another

Bridgeland stability condition. In fact, we have S†i = ρ−1(S′−i) where S′n : n ∈ Z is the

set of new special Lagrangian spheres in Xs=1 which map to straight line segments by

z-projection as in Figure 1.10.

Figure 1.10: Special Lagrangian spheres S′k in Xs=1.

For later use we orient these spheres as follows. In conic fiber direction, each Si restricts

to a 2-dimensional torus |u1| = |v1|, |u2| = |v2|. We fix the orientation on the fiber torus

to be dθ1 ∧ dθ2 where θi are the arguments of ui respectively. We orient their matching paths

as in the right side of Figure 1.9.

Set L0 := S0 and L1 := S1. They are distinguished objects in the set Sn of Lagrangian

spheres in the sense that they have minimal/maximal slopes (or phases) as well as they

generate DbF . We will study Lagrangian Floer theory of L0 and L1 intensely in Section

1.6.1.

36

Recall from Section 1.2 that DY/Y is the subcategory of Db(Y) generated by OC(−1)[1]

and OC. [21] proved the following equivalence of subcategories of Fuk(Xs=0) and Db(Y).

Theorem 1.5.3. ([21]) There is an equivalence DbF ' DY/Y, sending

L0 7→ OC(−1)[1] L1 7→ OC. (1.7)

Using the chain model of Abouzaid [1], they explicitly computed the A∞-structure of

the endomorphism algebras of L0 ⊕ L1 to conclude that

End(L0 ⊕ L1) ' End(OC(−1)[1]⊕OC). (1.8)

See [21, Section 5, 7] for more details.

In this chapter, we shall use either the Morse–Bott model in [38] or pearl trajectories

[11, 90] to study Lagrangian torus fibers and the noncommutative mirror functor. They are

conceptually easier to understand.

Figure 1.11: Transformation of L0 and L1 by the symplectomorphism ρ.

The A-flop can be realized by the symplectomorphism ρ from Xs=0 to Xs=1 given in

1.4.4 (see Figure 1.7). Figure 1.11 shows how ρ acts on Li, where the third diagram describes

the moment at which L0 and L1 happen to have the same phases. Observe that Xs=1 (1.6)

is obtained from Xs=0 by swapping two sets of coordinates (u1, v1) and (u2, v2). However,

swapping the coordinates is different from the symplectomorphism that gives A-flop, as its

effect on z-plane shows.

37

As in Figure 1.11, ρ sends L0 and L1 to special Lagrangian spheres in Xs=1 which we

denote by L′0 and L′1 respectively. Let F ′ denote the Fukaya subcategory of Xs=1 consisting

of L′0 and L′1. There is a natural functor ρ∗ : F → F ′ induced by the symplectomorphism ρ.

On the other hand, we can repeat the same argument as in the proof of Theorem 1.5.3 to see

that

DbF ′ ' DY†/Y with L′0 7→ OC†(−1) and L′1 7→ OC†(−2)[1].

Notice that this identification is coherent with the fact that L′0 is somewhat similar to the

orientation reversal of L0, whereas L1ρ7→ L′1 can be understood as a change of winding

number with respect to z = 0.

In fact, we have

End(L′0 ⊕ L′1) ' End(L0 ⊕ L1) (1.9)

as two set of objects are related by a symplectomorphism, and

End (OC†(−1)⊕OC†(−2)[1]) ' End (OC(−1)[1]⊕OC) (1.10)

due to the flop functor (see 1.2.2). It directly implies that the functor ρ∗ induced by the

symplectomorphism is mirror to the flop functor through the identification of A and B side

categories via [21]. Namely,

Proposition 1.5.4. We have a commutative diagram of equivalences:

DbF

ρ∗

' // DY/Y

Φ

DbF ′ ' // DY†/Y

(1.11)

Proof. It obviously commutes on the level of objects by the construction. (1.8), (1.9) and

(1.10) imply that the diagram also commutes on morphism level.

We shall study how the symplectomorphism ρ or its induced functor ρ∗ acts on various

geometric objects in DbF . For that, we should examine what kind of geometric objects are

actually contained in DbF .

38

1.5.1 Geometric objects of DbF

Let us first prove that any torus fiber intersecting L0 and L1 is isomorphic to a mapping

cone Cone(

L0α→ L1

)for some degree-1 element α ∈ HF(L0, L1) in the derived Fukaya

category. In particular this will imply that the category DbF contains those torus fibers as

objects.

One can choose the gradings on Li such that HF∗(L0, L1) = H∗(S1a)[−1]⊕ H∗(S1

b)[−1].

Here, we use the Morse–Bott model, where S1a and S1

b denotes the intersection loci over

z = a and z = b, respectively. Both S1a and S1

b are isomorphic to a circle. Thus degree 1

elements in HF(L0, L1) are given by linear combinations of (Poincarè duals of) fundamental

classes of S1a and S1

b. The cone Cone(L0α→ L1) can be identified with a boundary deformed

object (L0 ⊕ L1, α) (see [38] or [86]).

Let Lc := (for a < c < b) be a Lagrangian torus that intersects L0 at z = c. This condition

determines Lc uniquely, as components of Lc in double conic fiber direction should satisfy

the same equation as those of L0 and L1. We orient Lc as in Figure 1.9 in z-plane components,

and use the standard one (from dθ1dθ2 as for Si) along the conic fiber directions. Lc cleanly

intersects L0 and L1 along 2-dimensional tori which we denote by T0 := Lc ∩ L0 and

T1 := Lc ∩ L1. One can see that CF(Lc, L0) = C∗(T0) and CF(L1, Lc) = C∗(T1) for suitable

choice of a grading on Lc. Similarly, CF(L0, Lc) = C∗(T0)[−1] and CF(Lc, L1) = C∗(T1)[−1].

Let Uρ1,ρ2,ρz be a unitary flat line bundle on Lc whose holonomies along circles in the

double conic fibers are ρ1 and ρ2 and that along the circle in z-plane is ρz.

Lemma 1.5.5. If (ρ1, ρ2) 6= (1, 1), then

HF(L0, (Lc, Uρ1,ρ2,ρz)) = 0, HF(L1, (Lc, Uρ1,ρ2,ρz)) = 0. (1.12)

Proof. One can simply use the Morse–Bott model for each of cohomology groups in (1.12).

Each of this group is simply a singular cohomology of the intersection loci, equipped with

twisted differential. Since the intersection loci are 2-dimensional torus in the double conic

fiber, the twisting is determined by (ρ1, ρ2). Here we only have classical differential, as

there is no holomorphic strip between Li and Tc. One can easily check that the cohomology

39

vanishes if the twisting is nontrivial.

Alternatively, one can perturb Lagrangians to have transversal intersections as in Figure

1.12 to see that the Floer differential has coefficients ρ1 − 1 and ρ2 − 1, which are nonzero

for nontrivial (ρ1, ρ2).

Figure 1.12: Perturbation of Lc, Li and polygons contributing to mb1 on CF(L, (Lc, ρ)).

The lemma implies that (Lc, Uρ1,ρ2,ρz) has no Floer theoretic intersection with L0 or L1

unless ρ1 = ρ2 = 1. From now on, we will only consider flat line bundles of the type U0,0,ρz

on Lagrangian torus fibers, which will be written as Uρz instead of U0,0,ρz for notational

simplicity. Let P0 := PD[T0] ∈ CF∗(Lc, L0) = C∗(T0) and P1 := PD[T1] ∈ CF∗(L1, Lc) =

C∗(T1). We also set αa := PD[S1a ] ∈ CF∗(L0, L1) and αb := PD[S1

b] ∈ CF∗(L0, L1). Notice

that degαa = degαb = 1 whereas degP0 = degP1 = 0.

Lemma 1.5.6. P0 ∈ CF0((Lc, Uρz), (L0⊕L1, α)) and P1 ∈ CF0((L0⊕L1, α), (Lc, Uρz)) are cycles

with respect to m0,α1 and mα,0

1 respectively if and only if α is given as λaαa + λbαb ∈ CF1(L0, L1)

with (λa : λb) = (Tω(∆2)ρz : Tω(∆1))3 where ∆1 and ∆2 are triangles shaded in Figure 1.13.

3It is harmless to put T = e−1 since only finitely many polygons contribute to A∞-structures. Neverthelesswe will keep the notation T to highlight contributions from nontrivial holomorphic polygons.

40

Figure 1.13: Two triangles contributing to m1 and m2.

Proof. We will prove the lemma for P1 only, and the proof for P0 is similar. We pick a point ×

as in Figure 1.13 for representative of Uρz so that when boundary of a holomorphic polygon

passes this point, the corresponding mk-operation is multiplied by ρ±1z depending on the

orientation. (More precisely, the point × represent 2-dimensional subtorus in Lc lying over

this point, which is called a hyper-torus and used to fix the gauge of a flat line bundle in

[25].)

Observe that two holomorphic triangles ∆1 and ∆2 shown in Figure 1.13 contribute to

the following operations:

m2(λaαa, P1) = λaTω(∆1)P0, m2(λbαb, P1) = −ρzλbTω(∆2)P0. (1.13)

where P0 is PD[T0] regarded as an element of CF(L0, Lc) = C∗(T0)[−1] ⊂ CF((L0 ⊕

L1, α), (Lc, Uρz)) (note that degP0 = 1). We do not provide the precise sign rule here

since it is not crucial in our argument. Indeed we can assume that two operations in (1.13)

produces outputs with the opposite signs by replacing λb to −λb if necessary.

Therefore we see that

mα,01 (P1) = ∑

kmk(α, · · · , α, P1) =

(λaTω(∆1) − ρzλbTω(∆2)

)P0 = 0

if and only if λa and λb have the ratio as given in the statement.

41

We next prove that P0 and P1 in Lemma 1.5.6 give isomorphisms between two objects

(Lc, Uρz) and Cone(L0α→ L1) where α is chosen as in Lemma 1.5.6. Here, it is enough to

present the ratio between λa and λb, as the mapping cone does not depend on the scaling of

α by an element in C× (or in Λ \ 0 if we do not substitute T by e−1).

Proposition 1.5.7. We have (Lc, Uρz)∼= Cone(L0

α→ L1) in the derived Fukaya category of Xs=0

where α = λaαa + λbαb is chosen as in Lemma 1.5.6.

Proof. Let us fix λa and λb to be ρzTω(∆2) and Tω(∆1) for simplicity. We claim that

mα,0,α2 (P1, P0) = C(1L0 + 1L1), m0,α,0

2 (P0, P1) = C1Lc

for some common constant C. (One should rescale P0 and P1 to get strict identity morphisms.)

To see this, pick generic points “?" on L0, L1 as in Figure 1.13, whose number of appearance

in the boundary of holomorphic discs determines the coefficient of 1Li . The same triangles

in the proof of Lemma 1.5.6 now contribute as

mα,0,α2 (P1, P0) = λaTω(∆1)1L1 + ρzλbTω(∆2)1L0 = ρzTω(∆1)+ω(∆2) (1L0 + 1L1)

Here two contributions add up contrary to (1.13). In fact, the relative signs are completely

determined by z-directions since all the Lagrangians share the other directions, and one can

use the sign rule due to Seidel [87] for z-plane components.

It is easy to check that the computation does not depend on the choice of generic points

? (it is essentially because P0 and P1 are cycles). Likewise, ∆2 contributes to

m0,α,02 (P0, P1) = ρzλbTω(∆2)1Lc = ρzTω(∆1)+ω(∆2)1Lc .

In particular, Proposition 1.5.7 implies the following exact sequence in the derived

Fukaya category

L1 → (Lc, Uρz)→ L0[1]→ .

42

Remark 1.5.8. Analogously, the following gives an exact triangle in DY/Y:

OC → Oy → OC(−1)[1][1]→

for a point y in C, or equivalently Oy ∼= Cone(OC(−1)[1] → OC) for some degree 1 morphism.

Note that Oy is a SYZ mirror of one of torus fibers Lc (with a flat line bundle Uρz ). Proposition 1.5.7

and the above exact triangle shows that the equivalence (1.7) sends torus fibers to skyscraper sheaves

over points in C.

By symmetric argument (or by the triangulated structure on DbFuk(Xs=0)), one also has

Cone(L0[1]β→ Lc) ∼= L1

in DbFuk(Xs=0) where deg(β) = 1. Here, [1] can be though of as taking orientation reversal

of the z-component of L0 (or, more precisely, such change of grading). A similar statement

holds true for other Lagrangian spheres Sm : m ∈ Z (Figure 1.9).

Figure 1.14: Triangles contributing to m1 and m2 on CF(Sm[1]⊕ Sm+1, Lc) and CF(Lc, Sm[1]⊕ Sm+1).

Proposition 1.5.9. Lagrangian sphere Si for any i can be obtained from taking successive cones from

S0 and S1. More precisely, one has the following:

43

• for m ≥ 1,

Sm+1∼= Cone(Lc → Sm),

• for n ≤ 0,

Sn−1∼= Cone(Sn[1]→ Lc[1]).

Proof. We only prove the first identity, and the proof for the second can be performed in

a similar manner. One can easily check that Lc = Cone(Sm[1]α→ Sm+1) (for n ≤ 0) by the

same argument as in the proof of Lemma 1.5.6, where α is a degree 1 morphism from

Sn−1 to Sn[1]. The contributing pair of triangles are as shown in Figure 1.14, and hence we

should take into account the relative areas of these two triangles together with the location

of c ∈ (a, b), when we choose α. We omit the details as it is completely parallel to the proof

of Lemma 1.5.6.

From Lc = Cone(Sm[1]α→ Sm+1), we have an exact triangle in the derived Fukaya

category Sm+1 → Lc → Sm[1][1]→ or equivalently Sm → Sm+1 → Lc

[1]→, which implies

Sm+1 = Cone(Lc → Sm) for some degree 1 morphism.

We conclude that DbF contains a sequence of Lagrangian spheres Sn : n ∈ Z and

P1 \ 0, ∞-family of Lagrangian tori parametrized by (c, ρz), where two missing points 0

and ∞ are presumably corresponding to two singular torus fibers. Indeed, DbF contains

the cones Cone(L0λbαb→ L1), Cone(L0

λaαa→ L1), and we believe that they are isomorphic to two

singular fibers that pass through z = a and z = b, respectively. Although a similar argument

as in the proof of Proposition 1.5.7 seem to go through, we do not spell this out here due to

technical reasons.

Notice that the above Lagrangian submanifolds are all special, thus they are expected

to be stable objects in the Fukaya category. Later in Section 1.6, we will study their

transformations into noncommutative resolution of the conifold Y to give stable quiver

representations.

Remark 1.5.10. By the equivalence in Theorem 1.5.3, Lagrangians spheres correspond to line bundle

44

on the exceptional curve C (or their shifts) as follows:

Sm 7→ OC(m− 1) for m ≥ 1

Sn 7→ OC(n− 1)[1] for n ≤ 0.

One can easily check this comparing the cone relations in Proposition 1.5.9 and exact sequences

consisting of line bundles and skyscraper sheaves on C.

1.5.2 Mirror to perverse point sheaves

In this section, we describe how torus fibers (intersecting L0 and L1) are affected by

A-flop. We will see that they behave precisely in the same way as skyscraper sheaves

supported at points in C(⊂ Y). Note that points in Y are mirror to torus fibers in SYZ point

of view, and those in C are mirror to torus fibers that intersect Li. Thus it is natural to expect

that those torus fibers are transformed to unstable objects (i.e. non-special Lagrangians)

which can be written as mapping cones analogous to (1.3).

Figure 1.15: Transformation of torus fibers under ρ.

Proposition 1.5.11. The functor ρ∗ sends (Lc, ρz) to Cone(L′0α′→ L′1) for a degree 1 morphism α′.

45

Proof. The proof is the same as that of Lemma 1.5.6, except that α′ in HF(L′0, L′1) now

represent outer (non-convex) angle in the z-plane picture. Altenatively, since ρ∗ sends cones

to cones, and ρ∗(L0) = L′0, ρ∗(L1) = L′1, we have

ρ∗(Lc, ρz) = ρ∗(Cone(L0α→ L1)) = Cone(ρ(L0)

ρ∗(α)→ ρ(L1)) = Cone(L′0α′→ L′1)

where α′ is a degree one morphism from L′0 to L′1, and occupies outer angles (after z-

projection) as in Figure 1.15.

Remark 1.5.12. Note that we have an exact triangle

L′1 → ρ∗(Lc, ρz)→ L′0[1]→

from Proposition 1.5.11. Thus L′1 can be thought of as a subobject of ρ∗(Lc, ρz). L′1 has bigger phase

than ρ∗(Lc, ρz), which is another way to explain unstability of ρ∗(Lc, ρz).

Likewise, flop sends most of Lagrangians spheres in Sm : m ∈ Z to non-special objects.

In fact, it is easy to see from the picture that S0 and S1 are the only spheres in this family

that remain special after A-flop. We conclude that the equivalence ρ∗ does not preserve the

set of special Lagrangians, and hence the A-flop can be thought of as a nontrivial change of

holomorphic volume form, while keeping the symplectic structure as its induced from a

symplectomorphism. We will revisit this point of view in 1.6.6.

1.6 Non-commutative mirror functor for the deformed conifold

and stability conditions

So far, we have studied A-flop on the smoothing Xs=0 (1.6) of the conifold mostly in SYZ

perspective comparing with its SYZ mirror, the resolve conifold (taken away a divisor) . In

this section we will consider a certain quiver algebra as another mirror to Xs=0, which is

well-known to be a noncommutative crepant resolution of the conifold. The relation between

46

noncommutative resolution and commutative one will be explained later (see Remark 1.6.3).

The mirror quiver category will enable us to study stability conditions more explicitly.

There is a natural way to obtain the above quiver as a formal deformation space

of the object L = L0 ⊕ L1 (recall L0 = S0 and L1 = S1 are Lagrangian spheres with

maximal/minimal phases) in the Fukaya category. By the result in [24], such a construction

comes with an A∞-functor from a Fukaya category to the category of quiver representations.

We will construct geometric stability conditions using the functor, and examine A-flop on

these stability conditions.

We begin with an explicit computation of the A∞-structure on CF(L, L), which is crucial

to describe formal deformation space of L.

1.6.1 Floer cohomology of L

As our Lagrangian L is given as a direct sum, CF(L, L) consists of four components:

CF(L, L) = CF(L0, L0)⊕ CF(L1, L1)⊕ CF(L0, L1)⊕ CF(L1, L0).

The first two components are both isomorphic to the cohomology of the three-sphere as

graded vector spaces, and hence have degree-0 and degree-3 elements only. These elements

will not be used for formal deformations. We only take degree-1 elements for deformations,

so that the Z-grading is preserved.

Recall that L0 and L1 intersect along two disjoint circles. There are several computable

models for CF(L0, L1) and CF(L1, L0) provided in [1]. Explicit computation was given in

[21] using one of these models, which we spell out here.

Theorem 1.6.1. (See [21, Theorem 7.1].) The A∞-structure on CF(L, L) are given as follows. As

vector spaces,

CF(Li, Li) = Λ〈1Li〉 ⊕Λ〈[pt]Li〉 for i = 0, 1

CF(L0, L1) = Λ〈X〉 ⊕Λ〈Z〉 ⊕Λ〈Y〉 ⊕Λ〈W〉

CF(L1, L0) = Λ〈Y〉 ⊕Λ〈W〉 ⊕Λ〈X〉 ⊕Λ〈Z〉.

47

with degrees of generators given as

deg1Li = 0, deg[pt]Li = 3,

degX = degY = degZ = degW = 1,

degX = degY = degZ = degW = 2.

We have m1 ≡ 0 and m≥4 ≡ 0. The only nontrivial operations are

−m2(X, X) = −m2(Z, Z) = m2(Y, Y) = m2(W, W) = [pt]L0 ,

m2(X, X) = m2(Z, Z) = −m2(Y, Y) = −m2(W, W) = [pt]L1 ,

m3(X, Y, Z) = −m3(Z, Y, X) = W, m3(Y, Z, W) = −m3(W, Z, Y) = X,

m3(Z, W, X) = −m3(X, W, Z) = Y, m3(W, X, Y) = −m3(Y, X, W) = Z.

and those determined by the property of the unit 1Li .

In what follows, we take an alternative way to compute A∞-structure on CF(L, L)

hiring pearl trajectories, which is more geometric in the sense that it shows explicitly the

holomorphic disks (attached with Morse trajectories) contributing to the A∞-operations.

This will also help us to have geometric understanding of various computations to be made

later, although most of the proof will rely on algebraic arguments.

First we choose a generic Morse function fi on Li with minimum and maximum only for

i = 0, 1. We denote these critical points by 1Li , [pt]Li by obvious analogy, where deg1Li = 0

and deg [pt]Li = 3. Then CF(Li, Li) is defined to be the Morse complex of fi, which is

nothing but the 2-dimensional vector space generated by 1Li , [pt]Li .

The Morse trajectories of fi are described as follows. Recall that the two Lagrangians

L0 and L1 intersect along two disjoint circles which we denoted by S1a and S1

b (lying over

z = a and z = b, respectively). We want to argue that generically, there is a unique gradient

flow line in each Li that runs from S1a to S1

b and vice versa. We may assume that there are

no critical points on S1a or S1

b by genericity.

Let W−(S1

b

)be the unstable manifold of S1

b with respect to the Morse function fi on Li,

namely

W−(

S1b

)=

x ∈ L0 : ϕt(x) ∈ S1b for some t ≥ 0

48

where ϕt is the flow of ∇ fi. Including the maximum (1Li in our notation), the unstable

manifold of S1b is topologically a disk that bounds S1

b. Now observe that two circles S1a and

S1b form a Hopf link in Li. Therefore, generically S1

a intersects the disk W−(S1

b

)∪ max at

one point as shown in Figure 1.16. Therefore we see that there is a unique trajectory flowing

from S1a to S1

b. This is the only property of the Morse functions f0 and f1 which we will use

later.

Figure 1.16: A trajectory of the Morse function fi from S1a to S1

b .

For the other components of CF(L, L), we perturb L1 in double conic fiber direction as

in Figure 1.17, so that L0 and L1 intersect each other transversely at four different points

after perturbation. Therefore, both CF(L0, L1) and CF(L1, L0) are generated by these four

points, which we denote as follows.

CF(L0, L1) = Λ〈X〉 ⊕Λ〈Z〉 ⊕Λ〈Y〉 ⊕Λ〈W〉

CF(L1, L0) = Λ〈Y〉 ⊕Λ〈W〉 ⊕Λ〈X〉 ⊕Λ〈Z〉

with degrees of generators given as

degX = degY = degZ = degW = 1, degX = degY = degZ = degW = 2.

Here, X and X are represented by the same point, but regarded as elements in CF(L0, L1)

and CF(L1, L0) respectively, and similar for Y, Z, W. In fact, they can be thought of as

49

Poincare dual to each other. Obviously, they are all cycles (i.e. m1-closed) since opposite

strips (pairs of strips on cylinders in Figure 1.17) cancel pairwise. Therefore CF(L, L) comes

with a trivial differential.

Now we are ready to spell out A∞-algebra structure on CF(L, L) in terms of the above

model. Recall from [11, 90] that A∞-operation counts the configurations which consist of

several holomorphic disks (pearls) joined by gradient trajectories as shown in Figure 1.17.

The constant disk at X (and X) attached with flows to [pt]Li contributes as m2(X, X) = [pt]L0 .

Similarly, we have

m2(X, X) = m2(Z, Z) = −m2(Y, Y) = −m2(W, W) = −[pt]L0 ,

m2(X, X) = m2(Z, Z) = −m2(Y, Y) = −m2(W, W) = [pt]L1 .

Other m2’s are either determined by properties of units 1Li , or zero by degree reason.

Figure 1.17: Pearl trajectory contributing to m3(X, Y, Z).

Computation of m3 involves more complicated pearl trajectories. We give an explicit

picture for one of those trajectories, and the rest can be easily found in a similar way. In

Figure 1.17, one can see a pearl trajectory consisting of two bigons connected by a gradient

flow, which contributes to m3(X, Y, Z) with output W. The red colored connecting flow in

Figure 1.17 is precisely the gradient trajectory in Figure 1.16, and hence the corresponding

moduli is isolated. The pearl trajectory degenerates into a Morse tree when perturbing L0

50

back to the original position (see Figure 1.18), which one of models in [1] takes into account.

Figure 1.18: Morse trees for m3(X, Y, Z).

Consequently, we have the following complete list of nontrivial m3-operations:

m3(X, Y, Z) = −m3(Z, Y, X) = W, m3(Y, Z, W) = −m3(W, Z, Y) = X,

m3(Z, W, X) = −m3(X, W, Z) = Y, m3(W, X, Y) = −m3(Y, X, W) = Z.

We remark that the symplectic area of a pearl trajectory becomes zero after taking limit

back to the original clean intersection situation (so that it degenerates into a Morse tree),

which explains why there are no T appearing in the above computations. If one wants to

keep working with the perturbed picture, one can simply rescale generators so that the

coefficients of m3 to be still 1.

Note that the A∞-structure computed in this way precisely coincides with the one given

in Theorem 1.6.1.

1.6.2 Construction of mirror from L

As in [23], we take formal variables x, y, z, w and consider the deformation parameter

b = xX + yY + zZ + wW, and solve the following Maurer–Cartan equation:

m1(b) + m2(b, b) + m3(b, b, b) + · · · = 0. (1.14)

51

where mk with inputs involving x, y, z, w are defined simply by pulling out the coefficient of

Floer generators to the front, i.e.,

mk(x1X1, · · · , xkXk) = (−1)∗xkxk−1 · · · x1mk(X1, · · · , Xk).

Here (−1)∗ is determined by usual Koszul sign convention, and in particular, is positive

when xiXi is one of xX, yY, zZ, wW.

Remark 1.6.2. In [23], “weak" Maurer–Cartan equations were mainly considered (i.e. the right

hand side of (1.14) replaced by λ · 1L), but in our example, weak Maurer–Cartan equation cannot

have solutions unless λ = 0 due to degree reason.

By Theorem 1.6.1, the Maurer–Cartan equation (1.14) is equivalent to

(zyx− xyz)W + (wzy− yzw)X + (xwz− zwx)Y + (yxw− wxy)Z = 0.

Therefore, b is a solution of 1.14 if and only if (x, y, z, w) is taken from the path algebra

(modulo relations) A of the following quiver with the potential:

Q : v0·z

**

x

""·v1

yjj

w

cc Φ = (xyzw)cyc − (wzyx)cyc

where the vertex vi corresponds to the object Li and arrows correspond to degree 1 mor-

phisms between Li’s (or their associated formal variables). See [23, Section 6] for more

details. The quiver algebra has the following presentation:

A :=ΓQ

〈∂xΦ, ∂yΦ, ∂zΦ, ∂wΦ〉 =ΓQ

〈xyz− zyx, yzw− zwy, zwx− xwz, wxy− yxw〉 . (1.15)

Remark 1.6.3. (Q, Φ) is well-known to be the noncommutative crepant resolution of the conifold

(see for instance [27]). In fact, one can easily check that the subalgebra of A consisting of loops

based at one of vertices is isomorphic to the function algebra of the conifold. For e.g., if we set

α = xy, β = xw, γ = zy, δ = zw, then the relations among x, y, z, w force α, β, γ, δ to commute

with each other and to satisfy αδ = βγ.

52

By [23], we have an (trianglulated) A∞-functor from the Fukaya category to DbModA

(here, DbModA denotes dg-enhanced triangulated category)

ΨL : DbF → DbModA.

In the rest of the section, we will use the degree shift Ψ := ΨL[3] instead of ΨL itself, in

order to have the images of the geometric objects in DbF lying in the standard heart ModA

of DbModA. This point will be clearer after the computation of Ψ(Si) in the next section.

We will also see that the functor is an equivalence onto a certain subcategory of DbModA.

1.6.3 Transformation of L0 and L1 and their central charges

We first compute the images of L0 and L1 themselves under Ψ. Ψ(L0) is simply a chain

complex over A given by CF(L, L0), which is a direct sum

CF(L, L0) = CF(L0, L0)⊕ CF(L1, L0)

= 〈1L0 , [pt]L0〉 ⊕ 〈Y, W, X, Z〉

with a differential mb1. Recall that mb

1(p) = ∑k mk(b, · · · , b, p).

We have already made essential computations for mb1 (see Theorem 1.6.1). Using the

previous computation, the list of mb1 acting on the generators is given as follows:

mb1(1L0) = yY + wW

mb1(Y) = xwZ− zwX

mb1(W) = zyX− xyZ

mb1(X) = x[pt]L0

mb1(Z) = z[pt]L0

mb1([pt]L0) = 0

Therefore [pt]L0 is the only nontrivial class in the cohomology. Moreover, x[pt]L0 and

z[pt]L0 are zero in the cohomology, and hence we obtain a finite dimensional representation

of (Q, Φ) over C. Consequently, mb1-cohomology of Ψ(L0) ∈ DbModA is 1-dimensional

vector space (over C) supported over the vertex v0, generated by the class [pt]L0 . We remark

53

that this vector space sits in degree 0 part due to the shift Ψ = ΨL[3].

Likewise, Ψ(L1) gives a 1-dimensional C-vector space supported over v1 after taking

mb1-cohomology. In particular, the image of Ψ lies in a subcategory DbmodA consisting of

objects with finite dimensional cohomology.

Theorem 1.6.4. Ψ : DbF → DbmodA is a fully faithful embedding, which sends L0 an L1 to their

corresponding vertex simples. Moreover, it is an equivalence onto DbnilmodA, the full subcategory

of DbmodA consisting of objects with nilpotent cohomologies.

Proof. As DbnilmodA is a full subcategory of DbmodA, it suffices to prove that

Ψ : DbF → DbnilmodA

is an equivalence. Since the image of generators L0 and Li are vertex simples which are

nilpotent over A, Ψ lands on DbnilmodA.

We prove that the morphism level functor on hom(Li, Lj) (i, j = 0, 1) induces iso-

morphisms of cohomology groups. Without loss of generality, it is enough to consider

hom(L0, L0) and hom(L0, L1). By [23, Theorem 6.10], we know that both

Ψ1 : hom(L0, L0)→ hom(Ψ(L0), Ψ(L0))

Ψ1 : hom(L0, L1)→ hom(Ψ(L0), Ψ(L1))

induce injective maps on the level of cohomology. Thus, it is enough to check that

dim Ext(Ψ(L0), Ψ(L0)) = 2 and dim Ext(Ψ(L0), Ψ(L1)) = 4.

On the other hand, it is known by [28] (see [93] also) that DbnilmodA is equivalent to DY/Y

(1.2) with vertex simples corresponding to OC and OC(−1)[1]. Therefore, the computation

of endomorphisms of OC ⊕OC(−1)[1] due to [21, Section 5] finishes the proof.

From now on, we take DbnilmodA to be the target category of Ψ.

Let zi be the central charge of Li, namely zi :=∫

LiΩ. We define a central charge on

quiver representations as follows. For a representation V := (V0, V1) of (Q, Φ) (with some

54

maps between V0 and V1 which we omitted),

Z(V) := z0 dim V0 + z1 dim V1. (1.16)

We define a dimension vector of a representation V by dim(V) := (dim V0, dim V1) ∈

Z2≥0 for later use. For general objects in DbmodA, Vi above should be replaced by the

corresponding cohomology.

Proposition 1.6.5. The object level functor Ψ0 : Obj(

DbF)→ Obj

(Db

nilmodA)

is a central

charge preserving map.

Proof. The statement is obviously true for L0 and L1 as they are mapped to modules with 1-

dimensional cohomology supported at the corresponding vertices. Since the central charges

on both sides are additive (i.e. they are the maps from the K-groups) and Ψ is a triangulated

functor, the statement directly follows.

In particular, special Lagrangians L0 and L1 are sent to simple and hence stable objects

on quiver side.

1.6.4 Stables on quiver side

Set ζi to be the argument of zi taken in (0, π] for i = 0, 1. Since L0 and L1 are special

Lagrangians, ζi is nothing but the phase of Li. According to our convention (see the

discussion below Theorem 1.5.2), we have ζ0 > ζ1.

Nagao and Nakajima used the following notion of stability for the abelian category

modA.

Definition 1.6.6. We define stability of quiver representations of (Q, Φ) as follows.

1. We define the phase function ζ on modA by

ζ(V) = ζ0 dim V0 + ζ1 dim V1 ∈ R

for V ∈ modA.

55

2. An object V of modA is said to be stable (semistable resp.) if for any subobject W of V,

ζ(W) < ζ(V) (ζ(W) ≤ ζ(V) resp.).

It is elementary to check that arg Z(V) ∈ (0, π] for V ∈ modA induces the equivalent

stability on the abelian category modA as Definition 1.6.6. In fact, one can easily check that

two quantities have the same ordering relations.

Lemma 1.6.7. For V, W ∈ modA, ζ(V) ≤ ζ(W) if and only if arg Z(V) ≤ arg Z(W).

The proof is elementary, and we omit here. In particular, the set of stable objects remains

the same even if we use arg Z(V) in place of ζ(V).

Remark 1.6.8. One advantage of using Z(V) (rather than ζ) is that it can be lifted to a Bridgeland

stability condition on DbnilmodA. In fact, one can check that it is equivalent to the perverse stability

on DY/Y via the identification DbnilmodA ∼= DY/Y. [74, Remark 4.6] gives a correspondence

between stable objects in the heart of each category.

We briefly review the classification of stable objects in modA following [74]. We first

set up the notation as follows. We define A-module V±(m) by (Cm, Cm±1) together with

the maps corresponding arrows given as in the left two columns in (1.17) where right (left,

resp.) arrows are x, z (y, w, resp.). Likewise, V†±(m) denotes A-module (Cm, Cm∓1) which

can be visualized as the right two columns in (1.17). Up to isomorphism, one can assume

that all arrows act by the identity map from C to itself.

56

C

C

33

++ C

C

33

......

C++ C

C

33

++ C

V+(m)

C** C

C

44

** C...

...

C

C**

44

C

C

44

V−(n)

CssC

C

kk

ssC...

...

C

C

kk

ssC

C

kk

V ′+(m)

C

C

jj

ttC

C

jj

......

CttC

C

jj

ttC

V ′−(n)(1.17)

We are now ready to state the classification result by Nagao-Nakajima.

Theorem 1.6.9. ([74, Theorem 4.5]) Stable modules W in modA are classified as follows:

1. when ζ0 > ζ1

• V+(m) (m ≥ 1)

• A-module W with dimW = (1, 1) parametrized by Y

• V−(n) (n ≥ 0)

2. when ζ0 < ζ1

• V ′+(m) (m ≥ 1)

• A-module W with dimW = (1, 1) parametrized by Y†

• V ′−(n) (n ≥ 0)

In particular, ζ1 = ζ2 gives a wall, and the wall structure consists of only two chambers.

In order to precisely match the pictures in [74], one should locate the vertex v1 to the left

in the quiver diagram. For instance, the vertex simple at the left vertex in [74] corresponds to

OC whereas in our case v1 (sitting on the right) represents the Lagrangian L1(= S1) which

is mirror to OC. See [74, Remark 4.6].

57

Remark 1.6.10. Modules with dimension vector (1, 1) in (1) and (2) of Theorem 1.6.9 can be

presented as

C

z))

x

""C

yii

w

aa (1.18)

for some (x, y, z, w) ∈ C4. Note that (1.18) is nilpotent if and only if either x = z = 0 or y = w = 0.

Notice that (1.18) has three dimensional deformation (scaling actions of arrows x, y, z, w up to overall

rescaling) whereas V±(k) is rigid for all k. Later, we will see that (1.18) is mirror to a Lagrangian

torus fibers whose first Betti number is 3, and V±(k) is mirror to a Lagrangian sphere.

In what follows, we shall show that transformations of special Lagrangians in DbF by

our functor Ψ recovers all the stable representations which are nilpotent.

1.6.5 Transformation of special Lagrangians in DbF before/after flop

We next compute the transformation of other geometric objects in DbF = Db〈L0, L1〉

under the mirror functor Ψ induced by L = L0 ⊕ L1. Recall from 1.5.1 that this category

contains Lagrangian spheres and torus fibers (intersecting spheres) as geometric objects.

We begin with a torus fiber Lc (a < c < b) in X that intersects each of L0 and L1 along

T2 at z = c. (see 1.5.1). Suppose Lc is also equipped with a flat line bundle U whose

holonomy along a circle in z-direction is ρ. Recall that we only consider U with trivial

holonomies along both of double conic fiber directions as otherwise they would not belong

to the category.

Lemma 1.6.11. The transformation of (Lc, ρ) by Ψ is the representation of (Q, Φ) (after taking

cohomology) given as

C λ1

//λ2 // C

v0·z

**

x

""·v1

yjj

w

cc

(1.19)

58

where [λ1 : λ2] parametrizes the exceptional curve C in Y. and the maps in the other direction (i.e.

actions of y and w) are zero.

Proof. Recall from Proposition 1.5.7 that (Lc, ρ) is a mapping cone Cone(L0α→ L1) for

nonzero α ∈ HF1(L0, L1) uniquely determined up to scaling. i.e. the following defines an

exact triangle in DbF

L1 → (Lc, ρz)→ L0[1]→ . (1.20)

Since the functor Ψ is a triangulated equivalence and α 6= 0, Ψ(Lc, ρz) is also a nontrivial

extension of Ψ(L1) and Ψ(L0). It is elementary exercise to show that all nontrivial extensions

of these two representations which are nilpotent should be of the form given in (1.19).

Moreover, since we have HF1(L0, L1) ∼= Ext1(Ψ(L0), Ψ(L1)) by the morphism level functor

of Ψ, we see that Ψ(Lc, ρ) gives all possible extensions as α varies, or equivalently c and ρ

(in (Lc, ρ)) vary. Note that the family of such (Lc, ρ) precisely parametrizes points in the

exceptional curve C by SYZ mirror construction due to [21].

We remark that the cones Cone(L0λaαa→ L1) and Cone(L0

λbαb→ L1) in DbF (which are

supposedly singular torus fibers) are sent to the representations of the same form with

one of λi being zero, which together with Ψ(Lc, ρ) completes the P1-family of stable

representations.

Geometric argument

We provide a more geometric computation of Ψ(Lc, ρ) making use of pearl trajectory

model introduced in 1.6.1. As shown in Figure 1.12, Lc intersect L0 ∪ L1 at eight different

points after perturbation. Let

L0 ∩ Lc := a00, a01, a10, a11

L1 ∩ Lc := b00, b01, b10, b11

(see Figure 1.12) where degaij = i + j + 1 and degbij = i + j. Note that the degrees for

generators in L0 ∩ Lc are shifted by 1 due to Floer theoretic grading from intersections in

z-direction.

59

mb1 for the above generators can be computed as follows:

mb1(b00) = (Tω(∆1)x− ρTω(∆2)z)a00 + xyb01 + zyb10

mb1(b01) = (ρTω(∆2)z− Tω(∆1)x)a01 − zwb11

mb1(b10) = (ρTω(∆2)z− Tω(∆1)x)a10 + xyb11

mb1(b11) = (Tω(∆1)x− ρTω(∆2)z)a11

mb1(a00) = yxa01 + wza10

mb1(a01) = −wza11

mb1(a10) = yxa11

mb1(a11) = 0

Here, the coefficients of the form (Tω(∆1)x− ρTω(∆2)z) in front of aij (in mb1(bij)) is from the

pair of holomorphic polygons projecting to the shaded triangles ∆1 and ∆2 in z-plane shown

in Figure 1.12. (They are the same triangles appearing in the proof of Lemma 1.5.6.) Other

terms are contributed by pearl trajectories consisting of two 2-gons joined by a gradient

flow, which have the same shape as the one contributing to m3 drawn in Figure 1.17.

Therefore the mb1-cohomology is generated by [a11] asA-module, and we have Tω(∆1)x[a11] =

ρTω(∆2)z[a11] (since their difference is mb1(b11)). Moreover, these are the only nontrivial scalar

multiplication since wz[a11] = yx[a11] = 0. In particular, we see that λ1, λ2 in (1.19) satisfy

λ1 : λ2 = ρTω(∆2) : Tω(∆1).

On the other hand, Ψ transforms Lagrangian spheres Sk into the following stable

representations.

Proposition 1.6.12. The images of spheres Sk : k ∈ Z in DbF (after taking cohomology) are

given as follows:

1. For m ≥ 1, Ψ(Sm) = V+(m),

2. For n ≤ 0, Ψ(Sn) = V−(|n|).

where V±(k) are as in (left two columns of) (1.17).

Proof. We will only prove (1), and the proof of (2) can be done in a similar manner. The

statement is true for m = 1 by Theorem 1.6.4. We will proceed by induction. Let us assume

that it is true for m. By Proposition 1.5.9, we have

Sm+1∼= Cone(Lc

α→ Sm)

60

for some α which implies the exact triangle Sm → Sm+1 → Lc[1]→ in DbF . Since α is a

nonzero element in the Floer cohomology, Sm+1 is a nontrivial extension of Sm and Lc. We

see that Ψ(Sm+1) is an extension of V+(m) and Ψ(Lc) = C λ1

//λ2 // C . For simplicity, we

choose a suitable Lc such that λ1 = −λ2, and hence, after rescaling two arrows act as id and

−id respectively.

On the other hand, we already know one nontrivial extension of these two modules,

which is nothing but V+(m + 1). To see this, observe that the map σ : V+(m)→ V+(m + 1)

defined by

σ :ei 7→ ei + ei+1,

f j 7→ f j + f j+1

is an injective A-module map, where ei (resp. f j) denotes the standard basis of V+(m)

spanning the i-th (resp. j-th) component C over v0 (resp. v1) for 1 ≤ i ≤ m − 1 (resp.

1 ≤ j ≤ m). See (1.21) below.

C = 〈 f1〉〈e1〉 = C

x 22

z ,,C = 〈 f2〉

〈e2〉 = C

22

......

〈em−2〉 = C,,C = 〈 fm−1〉

〈em−1〉 = C

x 22

z ,,C = 〈 fm〉

V+(m)

(1.21)

Now the cokernel of σ is spanned by [e1] and [ f1], and x[e1] = [xe1] = [ f1] and z[e1] =

[ze1] = [ f2] = −[ f1] since f1 + f2 is in the image of σ. Therefore, we have

0→ V+(m)σ→ V+(m + 1)→

(C −id

//id // C)→ 0.

61

Moreover, V+(m + 1) is the only nontrivial extension of V+(m) and C −id//id // C since

dim Ext(

C −id//id // C , V+(m)

)∼= dim HF1(Lc, Sm) = 1.

(Here, we used the fact that Ψ is an equivalence, and the induction hypothesis that Ψ(Sm) =

V+(m).) We conclude that Sm+1 should map to V+(m + 1) by Ψ.

Geometric argument

Alternatively, one can compute the image of spheres under Ψ directly by holomorphic

disk counting (or more precisely computing mb1 on CF(L, Si)). We give a brief sketch of

the computation for Sm for m ≥ 1. On z-plane, the projection of Sm intersects the interval

(a, b) (m− 1)-times (not including the end points a and b). Here, we perturb L0, L1 and Sm

along the fiber direction as in the proof of Lemma 1.6.11 so that they mutually intersect

transversely.

Let us denote these m− 1 points in (a, b) by c1, c2, · · · , cm−1 as shown in Figure 1.19.

These are the only locations where one has the highest degree intersections (i.e. degree 3

elements in CF(L, Sm) that map to degree 0 elements by Ψ = ΨL[3]). Cohomology long

exact sequence tells us that it is enough to consider these elements, as cohomologies of

Ψ(S0) and Ψ(S1) are supported only at this degree. We remark that the intersection L ∩ Sm

occurring at z = a and z = b only produces degree 0 and 1 elements in the Floer complex,

which are not in the highest degree.

Denote these highest degree intersection points by

c1, c2, · · · , cm−1

where ci projects down to ci. (Obviously there is only one highest degree intersection point

over each ci after perturbation.) There are pairs of triangles whose z-projections are given as

shaded region in Figure 1.19. Each of these pair contributes to mb1 with the same input, say

ri as in the figure, and gives rise to

mb1(ri) = zci − xci+1. (1.22)

62

Figure 1.19: Contributing holomorphic polygons to (the differential of) Ψ(Sm).

Also, there are pearl trajectories with a similar shape to the ones contributing to formal

deformation (m3) of L, which lies over the interval [a, b]. These trajectories induce

mb1(pi) = ±yxci, mb

1(qi) = ±wzci (1.23)

where pi and qi are the two degree 2 intersection points lying over ci.

Set ei := [ci] which belong to the v0-component of the resulting quiver representation .

(1.22) implies

zei = xei+1,

and we denote this element by fi which belongs to v1-component. We also set f1 := xe1 and

fm := zem−1. Combining (1.22) and (1.23) implies all other actions of A are trivial. Therefore

the resulting cohomology has precisely the same as V+(m) as a A-module.

Effect of A-flop

The images of L0 = S0 and L1 = S1 under the symplectomorphism ρ : Xs=0 → Xs=1

gives another Lagrangian spheres S′0 and S′1 (see Figure 1.11). In addition, we have a new

sequence of special Lagrangian spheres

S′k : k ∈ Z

depicted in Figure 1.10. Readers are

63

warned that S′k is not a image of Sk under A-flop unless k = 0 or k = 1. Note that phases of

other spheres lie between those of S′0 and S′1 due to our choice of gradings (or orientations)

as in Figure 1.10. (Recall that we measure the phase angles in clockwise direction.)

We perform the same mirror construction making use of L′0 = S′0 and L′1 = S′1 which

obviously produce the same quiver with the potential. Only difference is that now the

ordering of the phases of L′0 and L′1 are switched. Namely, in this case, we have ζ ′0 < ζ ′1

where ζ ′i is a phase of L′i. Thus one can naturally expect to obtain stable representations in

(2) of Theorem 1.6.9 by applying the resulting functor Ψ′ : Db〈S′0, S′1〉 → DbnilmodA to

S′k

and new torus fibers (which can be represented as straight vertical lines in Figure 1.10).

Proposition 1.6.13. Torus fibers intersecting S′0 and S′1 are transformed under Ψ′ into

C Cλ1ooλ2

oo

after taking mb1-cohomology, where [λ1 : λ2] parametrizes the exceptional curve C† in Y†. The images

of spheres S′k : k ∈ Z in DbF ′ (after taking cohomology) are given as follows:

1. For m ≥ 1, Ψ(S′m) = V†+(m),

2. For n ≤ 0, Ψ(S′n) = V†−(|n|).

where V†±(k) are as in (right two columns of) (1.17).

The proof is essentially the same as that of Proposition 1.6.12, and we will not repeat

here.

1.6.6 Bridgeland stability on DbF

As mentioned in Remark 1.6.8, DbnilmodA admits a Bridgeland stability through the iso-

morphism DbnilmodA ∼= DY/Y. Therefore, DbF also admits a Bridgeland stability condition

(Z,P) by pulling-back the one on DbnilmodA via the equivalence Ψ. By Proposition 1.6.5,

we see that the pull-back stability on DbF is geometric in the sense that its central charge

is given by the period∫

Ωs=0. Moreover, the discussion in 1.6.5 tells us that the special

Lagrangian spheres Sk and tori (that intersects spheres) are stable objects in the heart.

64

After applying A-flop, we consider the subcategory F ′ generated by S′0 and S′1 in Xs=1.

By the same reason, DbF ′ admits a Bridgeland stability condition whose stable objects

(in the heart) are special Lagrangian spheres S′k’s and new torus fibers in Xs=1. Note that

their inverse image under ρ : Xs=0 → Xs=1 are S†i , the A-flop of the spheres Si in Xs=0,

which were discussed in Section 1.4.3.

By pulling back this stability condition on DbF ′ via ρ∗, we get a new stability condition

(Z†,P†) on DbF (whose central charge comes from ρ∗Ωs=1) with the set of stable objects

P† = L† : L is stable with respect to (Z,P).

This proves Theorem 1.1.3.

65

Chapter 2

Mirror of Weil–Petersson metric1

2.1 Introduction

The present chapter investigates some differential geometric aspects of the space of

Bridgeland stability conditions on a Calabi–Yau triangulated category.

The motivation of our study comes from mirror symmetry. It is classically known that the

complex moduli spaceMcpx(Y) of a projective Calabi–Yau manifold Y comes equipped with

a canonical Kähler metric, called the Weil–Petersson metric. The existence of such a natural

metric often implies strong results that one cannot obtain by purely algebraic methods. In

the case of Calabi–Yau threefold, this metric provides a fundamental differential geometric

tool, known as the special Kähler geometry, to study mirror symmetry. In light of duality

between complex geometry and Kähler geometry of a mirror pair of Calabi–Yau manifolds,

a natural problem is to construct the mirror object of the Weil–Petersson geometry, which

should be defined on the stringy Kähler moduli space MKah(X) of a mirror Calabi–Yau

manifold X of Y. However, there is yet no mathematical definition ofMKah(X) at present.

In the celebrated work [14], Bridgeland introduced and studied stability conditions on a

triangulated category with the hope of rigorously definingMKah(X). He conjectured and

confirmed in several important cases [8, 14, 15, 16] that the string theorists’ stringy Kähler

1Co-authored with Atsushi Kanazawa and Shing-Tung Yau. Reference: [36].

66

moduli spaceMKah(X) admits an embedding into the double quotient

Aut(DX)\Stab(DX)/C

of the space Stab(DX) of Bridgeland stability conditions on the bounded derived category

DX = DbCoh(X) of coherent sheaves on X.

The purpose of this chapter is to provide a step toward differential geometric study

of MKah(X) via the space of Bridgeland stability conditions. Although we are mostly

interested in geometric situations, we will use a more general categorical language in

this chapter. On careful comparison of the two sides of Kontsevich’s homological mirror

symmetry DbCoh(X) ∼= DbFuk(Y), we will give a provisional definition of Weil–Petersson

geometry and propose a conjecture which refine a previously known one. We will also

provide some supporting evidence by computing basic geometric examples. A key example

is the following, where our Weil–Petersson metric coincides with the classical Bergman

metric on a Siegel modular variety (notations will be explained later).

Theorem 2.1.1 (Theorem 2.4.8, Corollary 2.4.12). Let A be the self-product Eτ × Eτ of an elliptic

curve Eτ. Then there is an identification

AutCY(DA)\Stab+N (DA)/C× ∼= Sp(4, Z)\H2.

Moreover, the Weil–Petersson metric on the stringy Kähler moduli space AutCY(DA)\Stab+N (DA)/C×

is identified with the Bergman metric on the Siegel modular variety Sp(4, Z)\H2.

This result is compatible with the mirror duality between A and the principally polarized

abelian surface. In fact, the complex moduli space of the latter is given by Sp(4, Z)\H2.

Another example is a quintic threefold X ⊂ P4. Assuming that a conjectural Bridgeland

stability condition exists, we will observe that the Weil–Petersson metric is given by a

quantum deformation of the Poincaré metric near the large volume limit.

It is worth noting that some aspects of the ideas in this chapter were presented in a

series of Wilson’s works [101, 104, 105] on metrics on the complexified Kähler cones. In

fact, his works are also motivated by mirror symmetry since the complexified Kähler cone

67

is expected to give a local chart of MKah(X) near a large volume limit. For example, in

the case of a Calabi–Yau 3-fold X, the curvature of the so-called asymptotic Weil–Petersson

metric was shown to be closely related to the trilinear form on H2(X, Z) [101, 105].

On the other hand, an advantage of our approach taken in this chapter is the fact that

our Weil–Petersson metric is global and makes perfect sense away from large volume limits,

in contrast to Wilson’s local study. As a matter of fact, the global aspects of the moduli

space are of special importance in recent study of mirror symmetry and string theory.

However, the real difficulty of the subject is to understand the precise relation between

the stringy Kähler moduli spaceMKah(X) and the space of Bridgeland stability conditions.

Nevertheless, as an application of our work, we find a new condition on the conjectural

embedding of MKah(X) into a quotient of the space of Bridgeland stability conditions,

namely the pullback of the Weil–Petersson metric onMKah(X) should be non-degenerate

(Conjecture 2.3.6). This simply means that the mirror identificationMKah(X) ∼=Mcpx(Y)

respects the Weil–Petersson geometries.

This chapter is organized as follows. In Section 2.2, we provide basic backgrounds on

the Bridgeland stability conditions and twisted Mukai pairings. In Section 2.3, after a brief

review and reformulation of the classical Weil–Petersson geometry, we translate it in the

context of Bridgeland stability conditions on a Calabi–Yau triangulated category. Section

2.4 is the heart of the chapter. We carry out detailed calculations for two classes of abelian

surfaces to justify our proposal. Lastly, we discuss the case of a quintic 3-fold, and comment

on further research directions.

We use the following notations and conventions throughout this chapter. For an abelian

group A, we denote by Atf the quotient of A by its torsion subgroup. We write AK := A⊗Z K

for a field K. ch(−) denotes the Chern character and TdX denotes the Todd class of X.

We write <(z) and =(z) for the real and imaginary parts of z respectively. A Calabi–Yau

manifold is a complex manifold whose canonical bundle is trivial. Throughout the chapter,

we work over complex numbers C.

68

2.2 Bridgeland stability conditions

2.2.1 Review: Bridgeland stability conditions

In the seminal work [14], Bridgeland introduced the notion of stability conditions on a

triangulated category D, which was motivated from M.R. Douglas’ work on Π-stability in

physics. We recall the definition and some basic properties of Bridgeland stability conditions

that will be needed in later discussions.

Throughout the article, we assume that D is essentially small, linear over C, and is of

finite type. The last condition means that for any pair of objects E, F ∈ D, the C-vector space

⊕i∈ZHomD(E, F[i]) is of finite-dimensional.

The Euler form χ on the Grothendieck group K(D) is given by the alternating sum

χ(E, F) := ∑i(−1)i dimC HomD(E, F[i]).

The numerical Grothendieck group N (D) := K(D)/K(D)⊥χ is defined to be the quotient

of K(D) by the null space of the Euler pairing χ.

We assume that D is numerically finite, that is, N (D) is of finite rank. One large class of

examples of such triangulated categories is provided by the bounded derived category of

coherent sheaves DbCoh(X) of a smooth projective variety X.

Now we recall the definition of Bridgeland stability conditions on a triangulated category

D. We will only consider the Bridgeland stability conditions that are full and numerical.

Definition 2.2.1 ([14]). A (full numerical) stability condition σ = (Z ,P) on a triangulated

category D consists of:

• a group homomorphism Z : N (D)→ C, and

• a collection of full additive subcategories P = P(φ)φ∈R of D,

such that:

1. If 0 6= E ∈ P(φ), then Z(E) ∈ R>0 · eiπφ.

69

2. P(φ + 1) = P(φ)[1].

3. If φ1 > φ2 and Ai ∈ P(φi), then Hom(A1, A2) = 0.

4. For every 0 6= E ∈ D, there exists a (unique) collection of distinguished triangles

0 = E0 // E1

// E2 //

· · · // Ek−1 // E

B1

cc

B2

``

Bk

aa

such that Bi ∈ P(φi) and φ1 > φ2 > · · · > φk. Denote φ+σ (E) := φ1 and φ−σ (E) := φk. The

mass of E is defined to be mσ(E) := ∑i |Z(Bi)|.

5. (Support property [66]) There exists a constant C > 0 and a norm || · || on N (D)⊗Z R such

that

||E|| ≤ C|Z(E)|

for any semistable object E.

The group homomorphism Z is called the central charge, and the nonzero objects in

P(φ) are called the semistable objects of phase φ. The additive subcategories P(φ) actually are

abelian, and the simple objects of P(φ) are said to be stable.

We denote Stab(D) the space of (full numerical) Bridgeland stability conditions on D.

There is a nice topology on Stab(D) introduced by Bridgeland, which is induced by the

generalized distance:

d(σ1, σ2) = sup0 6=E∈D

|φ−σ2

(E)− φ−σ1(E)|, |φ+

σ2(E)− φ+

σ1(E)|, | log

mσ2(E)mσ1(E)

|∈ [0, ∞].

The forgetful map

Stab(D) −→ Hom(N (D), C), σ = (Z ,P) 7→ Z

is a local homeomorphism [14, 66]. Hence Stab(D) is a complex manifold.

There are two important group actions on the space of Bridgeland stability conditions

Stab(D):

70

• The group of autoequivalences Aut(D) on the triangulated category D acts on Stab(D)

as isometries with respect to the generalized metric: Let Φ ∈ Aut(D) be an autoequiv-

alence,

σ = (Z ,P) 7→ Φ · σ := (Z [Φ]−1,P ′),

where [Φ] is the induced automorphism on N (D), and P ′(φ) := Φ(P(φ)).

• The abelian group of complex numbers C acts freely on Stab(D): Let z = x + iy ∈ C

be a complex number,

σ = (Z ,P) 7→ z · σ := (ezZ ,P ′′),

where P ′′(φ) := P(φ− yπ ).

These two actions commute with each other. Note that there actually is a larger group

C ⊂ ˜GL+(2, R) acting on the space of stability conditions Stab(D), which “shears" the

central charges and also commutes with the Aut(D)-actions.

Remark 2.2.2. Let DbFuk(Y) be the derived Fukaya category of a Calabi–Yau manifold Y. We fix a

holomorphic volume form Ω of Y. It is a folklore conjecture (c.f. [16]) that there exists a Bridgeland

stability condition on DbFuk(Y) with central charge given by the period integral

Z(L) =∫

LΩ.

Moreover, the special Lagrangian submanifolds of phase φ are the semistable objects of phase φ with

respect to this stability condition.

2.2.2 Central charge via twisted Mukai pairing

Let X be a smooth projective variety. Motivated by work of Mukai in the case of

K3 surfaces, Caldararu defined the the Mukai pairing on H∗(X; C) as follows [26]: for

v, v′ ∈ H∗(X; C),

〈v, v′〉Muk :=∫

Xec1(X)/2 · v∨ · v′.

71

Here v = ∑j vj ∈ ⊕jH j(X; C) and its Mukai dual v∨ = ∑j√−1

jvj ∈ H∗(X; C). Note that

the above Mukai paring differs from Mukai’s original one [73] for K3 surfaces by a sign. We

define a twisted Mukai vector of E ∈ DX = DbCoh(X) by

vΛ(E) := ch(E)√

TdX exp(√−1Λ)

for any Λ ∈ H∗(X; C) such that Λ∨ = −Λ. The usual Mukai vector is the special case

where Λ = 0. A twisted Mukai pairing is compatible with the Euler pairing; by the

Hirzebruch–Riemann–Roch theorem,

χ(E, F) =∫

Xch(E∨)ch(F)TdX = 〈vΛ(E), vΛ(F)〉Muk.

A geometric twisting Λ compatible with the integral structure on the quantum cohomology

was introduced by Iritani [53] and Katzarkov–Kontsevich–Pantev [59]. We shall use a

reformulation due to Halverson–Jockers–Lapan–Morrison [49] in the following. First let us

recall a familiar identity from complex analysis

z1− e−z = ez/2 z/2

sinh(z/2)= ez/2Γ(1 +

z2π√−1

)Γ(1− z2π√−1

),

where Γ(z) is the Gamma function. The power series in the LHS induces the Todd class

TdX. We then consider a square root of the Todd class by writing√z

1− zexp(√−1Λ(z)) = ez/4Γ(1 +

z2π√−1

),

and solve it for Λ(z), where z is real, as

Λ(z) = =(log Γ(1 +z

2π√−1

))

=γz2π

+ ∑j≥1

(−1)j ζ(2j + 1)2j + 1

( z2π

)2j

where γ is Euler’s constant. Since the constant term of Λ(z) is zero, we may use it to define

an additive characteristic class ΛX, called the log Gamma class. Note that Λ∨X = −ΛX as

72

only odd powers of z appear in Λ(z). In the Calabi–Yau case, we can explicitly write it as

ΛX = − ζ(3)(2π)3 c3(X) +

ζ(5)(2π)5 (c5(X)− c2(X)c3(X)) + . . .

For K3 and abelian surfaces, there is no effect of twisting as ΛX = 0. For Calabi–Yau 3-folds,

the modification is precisely given by the first term, which is familiar in period computations

in the B-model side. We define vX(E) to be the twisted Muaki vector of an object E ∈ DX

associated to the log Gamma class ΛX, i.e.

vX(E) := ch(E)√

TdX exp(√−1ΛX).

Let X be a projective Calabi–Yau manifold equipped with a complexified Kähler param-

eter

ω = B +√−1κ ∈ H2(X; C),

where κ is a Kähler class. Let also q = exp(2π√−1ω). We define exp∗(ω) by

exp∗(ω) := 1 + ω +12!

ω ∗ω +13!

ω ∗ω ∗ω + · · · .

where ∗ denotes the quantum product. It is conjectured (c.f. [54]) that near the large volume

limit, which means that∫

C =(ω) 0 for all effective curve C ⊂ X, there exists a Bridgeland

stability condition on DX with central charge of the form

Z(E) = − 〈exp∗(ω), vX(E)〉Muk . (2.1)

Then the asymptotic behavior of the above central charge near the large volume limit is

given by

Z(E) ∼ −∫

Xe−ωvX(E) + O(q). (2.2)

The existence of a Bridgeland stability condition with the asymptotic central charge given

by the leading term of the above expression has been proved in various important examples

including K3 surfaces and abelian surfaces [15], as well as abelian threefolds [10, 69].

73

2.3 Weil–Petersson geometry

2.3.1 Classical Weil–Petersson geometry

We review some basics of the classical Weil–Petersson geometry on the complex moduli

space Mcpx(Y) of a projective Calabi–Yau n-fold Y. In fact, such a (possibly degenerate)

metric can be defined on the complex moduli space of a polarized Kähler–Einstein manifold

by the Kodaira–Spencer theory, but we will use a period theoretic method for a Calabi–Yau

manifold (see for example [96]). It fits naturally into the framework of the Hodge theory

and gives us a connection to the Bridgeland stability conditions.

First, we consider the following vector bundle on the complex moduli spaceMcpx(Y)

H = Rnπ∗C⊗OMcpx(Y) −→Mcpx(Y).

It comes equipped with a natural Hodge filtration F∗H of weight n. By the Calabi–Yau

assumption, the first piece of the filtration defines a holomorphic line bundle FnH →

Mcpx(Y), which is called the vacuum bundle. For a nowhere vanishing local section Ω of

the vacuum bundle, the quantity

KcpxWP(z) := − log

((√−1)n2

∫Y

Ωz ∧Ωz

)defines a local smooth function Kcpx

WP, known as the Weil–Petersson potential, on the complex

moduli spaceMcpx(Y). Then the Weil–Petersson metric onMcpx(Y) is defined to be the

Hessian metric√−12 ∂∂Kcpx

WP. A fundamental fact is that the Hessian metric is non-degenerate

and provides a canonical Kähler metric on the complex moduli spaceMcpx(Y).

Let χ(L1, L2) = χ(HF∗(L1, L2)) be the Euler pairing on the derived Fukaya category

DbFuk(Y). The following identity is useful for computing the Weil–Petersson potential.

Proposition 2.3.1. Provided that there exist formal sums of Lagrangian submanifolds Li repre-

senting a basis of Hn(Y; Z)tf, we have

KcpxWP(z) = − log

((√−1)−n ∑

i,jχi,j

∫Li

Ωz

∫Lj

Ωz

), (2.3)

74

where (χi,j) = (χ(Li, Lj))−1 is the inverse matrix.

Proof. Let Ai be a basis of Hn(Y; Z)tf. We define (γi,j) = (Ai · Aj)−1 to be the inverse of

the intersection matrix. Then by expanding Ωz and Ωz by the dual basis of Ai, we can

rewrite the Weil–Petersson potential as

KcpxWP(z) = − log

((√−1)n2

∑i,j

γi,j∫

Ai

Ωz

∫Aj

Ωz

).

On the other hand, for Lagrangian submanifolds L1, L2 ⊂ Y, the identity

[L1] · [L2] = (√−1)n(n+1)χ(L1, L2),

is standard in the Lagrangian Floer theory (see for example [40, Section 4.3]). This completes

the proof.

An advantage of the new expression (Equation (2.3)) is that it is not only categorical but

also Hodge theoretic in the sense that it is written in terms of period integrals.

2.3.2 Weil–Petersson geometry on StabN (D)

Motivated by Remark 2.2.2 and Proposition 2.3.1, we shall propose a definition of Weil–

Petersson geometry on a suitable quotient of the space of Bridgeland stability conditions on

a Calabi–Yau triangulated category D of dimension n ∈N, i.e. for every pair of objects E

and F, there is a natural isomorphism

Hom∗D(E, F) ∼= Hom∗D(F, E[n])∨.

An important consequence is that the Euler form on N (D) is (skew-)symmetric if n is even

(odd).

Let Ei be a basis of the numerical Grothendieck group N (D). We define a bilinear

form b : Hom(N (D), C)⊗2 → C by

Z1 ⊗Z2 7→ b(Z1,Z2) := ∑i,j

χi,jZ1(Ei)Z2(Ej),

75

where (χi,j) := (χ(Ei, Ej))−1. It is an easy exercise to check that the bilinear form b is

independent of the choice of a basis.

Definition 2.3.2. We define the subset Stab+N (D) ⊂ StabN (D) by

Stab+N (D) := σ = (Z ,P) | b(Z ,Z) = 0, (

√−1)−nb(Z ,Z) > 0.

The first condition is vacuous when n is odd as the bilinear form b is skew-symmetric.

Such conditions have been studied in the case of K3 surfaces (a dual description via the

Mukai pairing) under the name of reduced stability conditions [15]. We note that Stab+N (D)

is an analogue of a period domain in the Hodge theory, and the natural free C-action on

StabN (D) preserves the subset Stab+N (D).

Definition 2.3.3. Let s = (Zσ,Pσ) be a local holomorphic section of the C-torsor Stab+N (D) →

Stab+N (D)/C, then

KWP(σ) := − log((√−1)−nb(Zσ,Zσ)

)defines a local smooth function on Stab+

N (D)/C. We call it the Weil–Petersson potential on

Stab+N (D)/C.

Proposition 2.3.4. The complex Hessian√−12 ∂∂KWP of the Weil–Petersson potential KWP does not

depend on the choice of a local section s. Moreover, it descends to the double quotient space

Aut(D)\Stab+N (D)/C

away from singular loci.

Proof. The first assertion is standard. The second assertion follows from the fact that autoe-

quivalences are compatible with the Euler pairing, and the induced actions on the numerical

Grothendieck group N (D) send a basis to another basis. Therefore the local sections which

are identified by elements of Aut(D) differ only by multiplying local holomorphic functions,

and thus the well-definedness of the metric follows from that of b.

The situation is particularly interesting when n is odd, as StabN (D) naturally carries

76

a holomorphic symplectic structure. Given a symplectic basis Ei, Fi of N (D), the skew-

symmetric bilinear form b : Hom(N (D), C)⊗2 → C is simply

Z1 ⊗Z2 7→∑i

(Z1(Fi)Z2(Ei)−Z1(Ei)Z2(Fi)

),

which provides a nowhere vanishing holomorphic 2-form on StabN (D).

Example 2.3.5. As a sanity check, we shall carry out the above construction for the derived category

DX = DbCoh(X) of an elliptic curve X. Since the action of ˜GL+(2, R) on StabN (DX) is free and

transitive [14, Theorem 9.1], we have

Stab+N (DX) = StabN (DX) ∼= ˜GL+(2, R) ∼= C×H,

as a complex manifold. Thus the double quotient is

Aut(DX)\Stab+N (DX)/C ∼= PSL(2, Z)\H.

This is indeed the Kähler moduli space of the elliptic curve X. The normalized central charge at

τ ∈H is given by

Z(E) = −deg(E) + τ · rank(E).

Hence the Weil–Petersson potential is

KWP(τ) = − log((√−1)−1(Z(Op)Z(OE)−Z(OE)Z(Op))

)= − log(=(τ))− log 2.

This is the Poincaré potential on H and it descends to the Kähler moduli space PSL(2, Z)\H.

2.3.3 Refining conjecture

Let X be a projective Calabi–Yau n-fold. Then StabN (DX) can be considered as an

extended version of the stringy Kähler moduli spaceMKah(X) [16, Section 7.1]. It is akin

to the big quantum cohomology rather than the small quantum cohomology in the sense

that the tangent space ofMKah(X) is H1,1(X) while that of StabN (DX) is ⊕pHp,p(X). It is

77

conjectured by Bridgeland [14] that there should exist an embedding ofMKah(X) into

Aut(DX)\StabN (DX)/C.

Note that when n is odd, the double quotient is a holomorphic contact space thanks to

the holomorphic symplectic structure on StabN (DX).

Motivated by mirror symmetry and classical Weil–Petersson geometry, especially the

fact that Weil–Petersson metric is non-degenerate on Mcpx(X), we can now propose the

following, which refines the previous conjecture.

Conjecture 2.3.6. There exists an embedding of the stringy Kähler moduli space

ι :MKah(X) → Aut(DX)\Stab+N (DX)/C.

The complex Hessian of the pullback ι∗KWP of the Weil–Petersson potential KWP defines a Kähler

metric on MKah(X), i.e. non-degenerate. Moreover, it is identified with the Weil–Petersson

metric on the complex moduli space Mcpx(Y) of a mirror manifold Y under the mirror map

MKah(X) ∼=Mcpx(Y). When n = 3, the image of MKah(X) is locally a holomorphic Legendre

variety.

We checked that the conjecture holds for the elliptic curves (Example 2.3.5) and will

provide more supporting evidence in the next section. It is worth noting that the real

difficulty lies in providing a mathematical definition of the stringy Kähler moduli space

MKah(X). One potential application of the above conjecture is that, we can make use of the

non-degeneracy condition on the Weil–Petersson metric to characterizeMKah(X).

2.4 Computation

We begin our discussion with Calabi–Yau surfaces, for which there is a mathematical

definition of stringy Kähler moduli space via the Bridgeland stability conditions [8, Section

7]. Our computation heavily relies on existing deep results, mainly due to Bridgeland, and

we do not claim originality. The purpose of this section is to back up our conjecture by

78

concrete examples.

2.4.1 Self-product of elliptic curve

We consider the self-product A := Eτ × Eτ of an elliptic curve Eτ := C/(Z + τZ) for

a generic τ ∈ H. We denote by NS(A) := H2(A, Z) ∩ H1,1(A) the Néron–Severi lattice

of A. Before considering the space of stability conditions, let us take a look at the set of

complexified Kähler forms ω ∈ NS(A)C. Let dz1 be a basis of H1,0(Eτ) of the first Eτ factor

of A, and dz2 similarly. Then a complexified Kähler form ω can be expressed as

ω =√−1 (ρdz1 ∧ dz1 + τdz2 ∧ dz2 + σ(dz1 ∧ dz2 − dz1 ∧ dz2))

such that the imaginary part =(ω) is a Kähler form. The real part <(ω) is often called a

B-field. Let Hg be the Siegel upper half-space of degree g defined by

Hg := M ∈ M(g, C) | Mt = M,=(M) > 0.

In this abelian surface example, we do not fix a polarization, but we vary it in a 3-dimensional

space H2 as follows.

Proposition 2.4.1 ([57]). Let Ag := E×gτ be the self-product of g copies of an elliptic curve Eτ The

set of complexified Kähler forms can be identified with the Siegel upper-half space Hg of genus g. In

the g = 2 case, the identification is given by the assignment ω 7→ Mω :=

ρ σ

σ τ

.

Proof. The g = 1 case is standard. Suppose that g = 2. It suffices to show that =(Mω) > 0.

Since ω is a complexified Kähler form, we have

tr(=(Mω)) = =(ρ) +=(τ) =∫

Eτ×pt=(ω) +

∫pt×Eτ

=(ω) > 0

and

det (=(Mω)) = =(ρ)=(τ)−=(σ)2 = =(ω)2 > 0,

and thus Mω ∈ H2.

79

On the other hand, since A2 contains no rational curves, the Kähler cone coincides with

the connected component of the positive cone in NS(S) which contains a Kähler class. This

readily proves the assertion for g = 2. Since we do not need the higher genus case, we leave

a proof to the reader (c.f. [57, Section 6]).

We now recall some notations in [15]. A result of Orlov [77, Proposition 3.5] shows that

every autoequivalence of DA induces a Hodge isometry of the lattice H∗(A; Z) equipped

with the Mukai pairing. Hence there is a group homomorphism

δ : Aut(DA) −→ AutH∗(A; Z).

The kernel of the homomorphism will be denoted by Aut0(DA).

Let Ω ∈ H2(A; C) be the class of a nonzero holomorphic two-form on A. The sublattice

N (A) := H∗(A; Z) ∩Ω⊥ ⊂ H∗(A; C)

can be identified with N (DA) = H0(A; Z)⊕NS(A)⊕ H4(A; Z) and has signature (3, 2). In

fact, since the complex moduli τ ∈H is generic, N (DA) ∼= U⊕2 ⊕ 〈2〉 as a lattice. Here U is

the hyperbolic lattice, and 〈2〉 denotes an integral lattice of rank 1 with the Gram matrix (2).

We define a subset P(A) ⊂ N (DA)C consisting of vectors f ∈ N (DA)C whose real

and imaginary parts span a negative definite 2-plane in N (DA)R. This subset has two

connected components. We denote by P+(A) the component containing vectors of the form

fω := exp(ω) for a complexified Kähler class ω ∈ NS(A)C.

Let us review some results on the space of Bridgeland stability conditions on algebraic

surfaces following [15]. The central charge of a numerical stability condition is of the form

Zf(E) = −〈f, vA(E)〉Muk = −〈f, ch(E)〉Muk

for some f ∈ N (DA)C. When f = fω for some complexified Kähler class ω, one can con-

struct a stability condition with central charge Zfωusing the tilting theory and Bogomolov

inequality. Moreover, such stability conditions are geometric in the sense that all skyscraper

sheaves are stable and of the same phase. We denote by Stab†(A) ⊂ Stab(A) the connected

80

component containing the set of geometric stability conditions. The following result on the

global structure of Stab†(A) is due to Bridgeland [15, Section 15].

Theorem 2.4.2 ([15]). Let A be an abelian surface over C.

1. The forgetful map π sending a stability condition to the associated vector f ∈ N (DA)C maps

onto the open subset P+(A) ⊂ N (DA)C. Moreover, the map

π : Stab†(A) −→ P+(A).

is the universal cover of P+(A) with the group of deck transformations generated by the double

shift functor [2] ∈ Aut(DA).

2. The action of Aut(DA) on Stab(A) preserves the connected component Stab†(A).

3. The group Aut0(DA) is generated by the double shift functor [2], together with twists by

elements of Pic0(A), and pullbacks by automorphisms of A acting trivially on H∗(A; Z).

Note that twists by elements of Pic0(A) and pullbacks by automorphisms of A acting trivially

on H∗(A; Z), act trivially on Stab†(A).

4. There exists a short exact sequence of groups

1 −→ Aut0(DA) −→ Aut(DA) −→ Aut+H∗(A; Z) −→ 1,

where Aut+H∗(A; Z) ⊂ AutH∗(A; Z) is the index 2 subgroup consisting of elements which

do not exchange the two components of P(A).

Following our proposal in the previous section, it is natural to consider the following

subset of Stab†(A).

Stab+N (A) := (Z ,P) ∈ Stab†(A) | b(Z ,Z) = 0, −b(Z ,Z) > 0

(c.f. Definition 2.3.2). Hence we need to compute the bilinear form b. We start with a lemma.

Lemma 2.4.3. Let X be a smooth projective variety of dimension n. Recall the twisted Mukai vector

vX(E) = ch(E)√

TdX exp(√−1ΛX). Then

81

1. Assume that X is Calabi–Yau, then 〈v, w〉Muk = (−1)n〈w, v〉Muk for any v, w ∈ H2∗(X; C).

2. Let Ei be a basis of the numerical Grothendieck group N (DX). Then

∑i,j〈v, vX(Ei)〉Muk · χi,j · 〈vX(Ej), w〉Muk = 〈v, w〉Muk,

where (χi,j) = (χ(Ei, Ej))−1.

Proof. The first assertion follows directly from the Serre duality. The second assertion is a

simple linear-algebraic fact which follows from the identity χ(Ei, Ej) = 〈vX(Ei), vX(Ej)〉Muk.

Note that Lemma 2.4.3 is purely algebraic and thus holds in a categorical setting as well.

We now compute the bilinear form b, with central charge of the form

Zf(E) = −〈f, vX(E)〉Muk,

where f ∈ H∗(X; C).

Lemma 2.4.4. b(Zf1 ,Zf2) = (−1)n〈f1,f2〉Muk.

Proof. This is a simple application of Lemma 2.4.3. We have

b(Zf1 ,Zf2) = ∑i,j

χi,j · 〈f1, vX(Ei)〉Muk · 〈f2, vX(Ej)〉Muk

= (−1)n ∑i,j

χi,j · 〈f1, vX(Ei)〉Muk · 〈vX(Ej),f2〉Muk

= (−1)n〈f1,f2〉Muk.

Using Lemma 2.4.4, we can determine which stability conditions lie in the subset

Stab+N (A) ⊂ StabN (A).

Proposition 2.4.5. A stability condition (Zf,P) ∈ Stab†(A) lies in Stab+N (A) if and only if

〈f,f〉 = 0 and −〈f,f〉 > 0. This condition is equivalent to that f is of the form fω = c exp(ω)

for some constant c ∈ C and complexified Kähler class ω.

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Proof. The first assertion follows directly from Lemma 2.4.4. The second assertion follows

from an explicit calculation of the Mukai pairing on surfaces.

Note that central charge of the form

Zfω(E) = −〈fω, vA(E)〉Muk

where fω = exp(ω) for a complexified Kähler class ω, has been discussed in physics

literatures, see Section 2.2.2. One can prove that a stability condition with such a central

charge always lies in Stab+N (D).

Proposition 2.4.6. 1. The central charge Zfωsatisfies

b(Zfω,Zfω

) = 0, (√−1)−nb(Zfω

,Zfω) > 0.

2. The Weil–Petersson potential at a stability condition with central charge Zfωis given by

KWP(ω) = − log(=(ω)n)− log(2n

n!

).

Proof. 1. By applying Lemma 2.4.4, we have

(√−1)−nb(Zfω

,Zfω) = (

√−1)n〈fω,fω〉Muk

=(√−1)n

n!(−ω + ω)n

=2n

n!=(ω)n > 0.

The last inequality is a consequence of the fact that ω is a complexified Kähler class.

The equality b(Zfω,Zfω

) = 0 also follows from the above by replacing Zfωby Zfω

.

2. The assertion follows from the above explicit computation.

Motivated by Proposition 2.4.5, we define the following subset R+(A) of P+(A).

R+(A) := f ∈ P+(A) | 〈f,f〉Muk = 0, −〈f,f〉Muk > 0.

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By Theorem 2.4.2 (1), the forgetful map Stab+N (DA)→ R+(A) is a covering map with the

group of deck transformation generated by the double shift functor [2] ∈ Aut(DA).

Lemma 2.4.7. R+(A)/C× ∼= H2 as a complex manifold. Thereby we have an identification

〈[2]〉\Stab+N (DA)/C× ∼= H2.

Proof. The quotient

R+(A)/C× ∼= Cf ∈ P(N (DA)C) | 〈f,f〉Muk = 0, −〈f,f〉Muk > 0

is the symmetric domain of type IV3. The assertion follows from the standard identification

IV3 ∼= III2 of the symmetric domains. More explicitly, it is given by the tube domain

realization H2 → R+(A)/C× : ω 7→ [fω].

We now recall the definition of Calabi–Yau autoequivalences following the work of Bayer

and Bridgeland [8]. Define

Aut+CYH∗(A) ⊂ Aut+H∗(A)

to be the subgroup of Hodge isometries which preserve the class of holomorphic 2-form

[Ω] ∈ PH∗(A; C). Any such isometry restricts to give an isometry of N (DA). In fact,

Aut+CYH∗(A) ⊂ AutN (DA)

is the subgroup of index two which do not exchange the two components of P(A).

An autoequivalence Φ ∈ Aut(DA) is said to be Calabi–Yau if the induced Hodge isometry

δ(Φ) lies in Aut+CYH∗(A). We denote AutCY(DA) ⊂ Aut(DA) the group of Calabi–Yau

autoequivalences. By Theorem 2.4.2 (4), there exists a short exact sequence

1 −→ Aut0(DA) −→ AutCY(DA) −→ Aut+CYH∗(A) −→ 1.

We write Aut0tri(DA) ⊂ Aut0(DA) for the subgroup generated by twists by elements of

Pic0(A) and pullbacks by automorphisms of A acting trivially on H∗(A; Z). Recall from

84

Theorem 2.4.2 (3) that Aut0tri(DA) acts trivially on Stab†(DA). We define

AutCY(DA) := AutCY(DA)/Aut0tri(DA).

Then AutCY(DA) acts on Stab+N (DA), and there is a short exact sequence

1 −→ 〈[2]〉 −→ AutCY(DA) −→ Aut+CYH∗(A) −→ 1.

Theorem 2.4.8. The covering map π induces an isomorphism

AutCY(DA)\Stab+N (DA)/C× ∼= Sp(4, Z)\H2

between the double quotient of Stab†N (DA) and the Siegel modular variety Sp(4, Z)\H2. We will

call it the stringy Kähler moduli space of A.

Proof. From the previous discussions, we have

AutCY(DA)\Stab+N (DA)/C× ∼= Aut+CYH∗(A)\H2.

The action of Aut+CYH∗(A) on H2 is purely lattice theoretic. As an abstract group Aut+CYH∗(A) ∼=

O+(U⊕2 ⊕ 〈2〉). By a fundamental result [43, Lemma 1.1] of Gritsenko and Nikulin, it can

be identified with the standard Sp(4, Z)-action on the Siegel upper-half space H2.

Remark 2.4.9. It is shown in [57] that Ag is mirror symmetric to a principally polarized abelian

surface of dimension g. Theorem 2.4.8 is thereby compatible with the fact that the Siegel modular

variety Sp(2g, Z)\Hg is precisely the complex moduli space of principally polarized abelian surfaces

of dimension g. For g > 2, we expect that Sp(2g, Z)\Hg is covered by a similar double quotient of a

suitable subset of Stab(DAg).

There exists a canonical metric on the Siegel modular variety Sp(4, Z)\H2, namely the

Bergman metric. It is known to be a complete Kähler–Einstein metric. The main theorem of

this section is to show that the Bergman metric coincides with the Weil–Petersson metric

defined by Definition 2.3.3.

Proposition 2.4.10 ([84]). The Bergman kernel KBer : Hg ×Hg → C of the Siegel upper half-space

85

Hg of degree g is given by

KBer(M, N) = −tr(log(−√−1(M− N))).

The Bergman metric is defined to be the complex Hessian of the Bergman potential

KBer(M) := KBer(M, M) = −tr(log(2=(M))).

Theorem 2.4.11. The Weil–Petersson potential on Stab+N (DA)/C coincides with the Bergman

potential of the Siegel upper half-plane H2 up to a constant.

Proof. By Proposition 2.4.5, the central charge of a stability condition in Stab+N (DA) is of the

form Z(E) = −c〈fω, vA(E)〉 for some complexified Kähler class ω. So we can apply the

calculation of the Weil–Petersson potential in Proposition 2.4.6.

The key idea is to use the identification of ω and Mω provided in Proposition 2.4.1. Then

the two Kähler potentials are related as follows:

KWP(ω) = − log(=(ω)2)− log 2

= − log(det(=(Mω)))− log 2

= −tr(log(2=(Mω))) + log 2

= KBer(Mω) + log 2.

This completes the proof.

Corollary 2.4.12. The Weil–Petersson metric on the stringy Kähler moduli space is identified with

the Bergman metric on Sp(4, Z)\H2 via the isomorphism in Theorem 2.4.8.

2.4.2 Split abelian surfaces

Now let A be a split abelian surface, that is, A ∼= Eτ1 × Eτ2 for elliptic curves Eτ1 and Eτ2 .

Such a splitting is unique provided that Eτ1 and Eτ2 are generic, or equivalently NS(A) ∼= U.

Discussions in the previous section carries over for the split abelian surface A. The set of

complexified Kähler forms is identified with H×H, which is diagonally embedded in H2

86

(c.f. Proposition 2.4.1). It is precisely the symmetric domain of type IV2 associated to U⊕2.

On the other hand, it is known (c.f. [50, Proposition 2.6]) that

Aut+CYH∗(A) ∼= O+(U⊕2) ∼= P(SL(2, Z)× SL(2, Z))o Z2,

where P(SL(2, Z)× SL(2, Z)) represents the quotient group of SL(2, Z)× SL(2, Z) by the in-

volution (A, B) 7→ (−A,−B) and the semi-direct product structure is given by the generator

of Z2 acting on SL(2, Z)× SL(2, Z) by exchanging the two factors.

Theorem 2.4.13. There is an identification

AutCY(DA)\Stab+N (DA)/C× ∼= P(SL(2, Z)× SL(2, Z))o Z2\(H×H)

Moreover, the Weil–Petersson metric on the stringy Kähler moduli space AutCY(DA)\Stab+N (DA)/C×

is identified with the Bergman metric on the Siegel modular variety P(SL(2, Z) × SL(2, Z))o

Z2\(H×H).

This observation is compatible with self-mirror symmetry for the split abelian surfaces.

In fact a lattice polarized version of the global Torelli Theorem asserts that the complex

moduli space of split abelian surfaces are given by the above Siegel modular variety.

Remark 2.4.14. A similar computation can be carried out for M-polarized K3 surfaces for the

lattice M ∼= U⊕2 ⊕ 〈−2〉 or U⊕2. The main difference is that there are spherical objects in the

derived category DX of a K3 surface X and we need to remove the union of certain hyperplanes from

P+. Moreover, the subgroup of Aut0(DX) which preserves the connected component Stab†(DX)

acts freely on Stab†(DX). So one does not need to take the quotient of the group of Calabi–Yau

autoequivalences by Aut0tri(D) as in the abelian surface case (c.f. [15]).

2.4.3 Abelian variety

Let X be an abelian variety of dimension n. Since there is no quantum corrections and

the Chern classes are trivial, the expected central charge at the complexified Kähler moduli

87

ω ∈ H2(X; C) is given by

Zfω(E) = −〈fω, vX(E)〉Muk = −

∫X

e−ωch(E).

The existence of Bridgeland stability condition with this central charge is known for n ≤ 3.

By Proposition 2.4.6, the Weil–Petersson potential is

KWP(τ) = − log(=(ω)n)− log2n

n!.

Fix a polarization H. We think of ω = τH for τ ∈ H as a slice of the stringy Kähler

moduli spaceMKah(X). Then the Weil–Petersson metric on H is essentially the Poincaré

metric. This example is a toy model in the sense that there is no quantum correction.

The above observation is compatible with Wang’s mirror result [102, Remark 1.3], which

says that in the case of infinite distance, the Weil–Petersson metric is asymptotic to a scaling

of the Poincaré metric.

2.4.4 Quintic threefold

Although the existence of a Bridgeland stability condition for a quintic threefold X ⊂ P4

has not yet been proven, we can still compute the Weil–Petersson potential using the central

charge in Equation (2.1) near the large volume limit.

Let τH ∈ H2(X; C) be the complexified Kähler class, where H is the hyperplane class

and τ ∈H. First we observe that

exp∗(τH) = 1 + τH +τ2

2(1 +

15 ∑

d≥1Ndd3qd)H2 +

τ3

6(1 +

15 ∑

d≥1Ndd3qd)H3,

where q = e2π√−1τ and NX

d denotes the genus 0 Gromov–Witten invariant of X of degree d,

and we use the quantum product

H ∗ H = Φ(q)H2 =15(5 + ∑

d≥1NX

d qdd3)H2.

88

Then the central charge computes to be

Z(E) = − 〈exp∗(τH), vX(E)〉Muk

= −∫

Xe−τHvX(E) +

ζ(3)χ(X)

(2π)3 (τ2

10H2ch1(E)− τ3

6ch0(E)) ∑

d≥1NX

d d3qd,

where χ(X) is the topological Euler number of X. Near the large volume limit, the Weil–

Petersson potential is given by

KWP(τ) = − log(

H3(Φ(q)(τ3

6+

ττ2

2)−Φ(q)(

τ3

6+

τ2τ

2))− 2 log

( ζ(3)χ(X)

(2π)3

)∼ − log(

43

H3=(τ)3)− 2 log( ζ(3)χ(X)

(2π)3

)+ O(q).

Therefore the Weil–Petersson metric of a quintic threefold is a quantum deformation of the

Poincaré metric on H as expected. In particular, for sufficiently small q, it is non-degenerate

and the Weil–Petersson distance to the large volume limit is infinite. When there is no

B-field, i.e. τ ∈√−1R, the correction term O(q) is explicitly given by log(Φ(q)).

Remark 2.4.15 ([18]). The stringy Kähler moduli spaceMKah(X) of a quintic Calabi–Yau threefold

X ⊂ P4 is expected to be identified with the suborbifold

[z ∈ C | z5 6= 1/Z5] ⊂ [P1/Z5].

The point z = ∞ is the large volume limit, the point z5 = 1 is the conifold point, and the point z = 0

is the Gepner point. We expect the following properties of the Weil–Petersson metric onMKah(X).

1. The Weil–Petersson distance to the conifold point, which corresponds to a quintic threefold with

a conifold singularity, should be finite. This is based on a result of Wang [102] on the mirror

complex moduli, which asserts that if a Calabi–Yau variety has at worst canonical singularities,

then it has finite Weil–Petersson metric along any smoothing to Calabi–Yau manifolds.

2. The Weil–Petersson metric at the Gepner point should be an orbifold metric. This is because

the auto-equivalence

Φ(−) = TOX ((−)⊗OX(H)

),

where TOX denotes the Seidel–Thomas spherical twist with respect to OX, at the Gepner

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point satisfies the relation Φ5 = [2]. This descends to Φ5 = id on K(DX). On other hand,

the calculations of Candelas–de la Ossa–Green–Parkes [18] shows that the Weil–Petersson

curvature tends to +∞ as we approach the Gepner point.

It is interesting to investigate the interplay among the geometry of a Calabi–Yau threefold

X, the cubic intersection form on H2(X; Z), and curvature properties of the Weil–Petersson

metric near a large volume limit [101, 103, 105].

On the other hand, probably a more alluring research direction is to examine the Weil–

Petersson metric away from a large volume limit, where central charges are not of the form

(2.1), as the metric is inherently global. For instance, the Weil–Petersson metric around a

Gepner point may be studied via matrix factorization categories via the Orlov equivalence

[78]

DbCoh(X) ∼= HMF(W),

where HMF(W) is the homotopy category of a graded matrix factorization of the defining

equation W of the quintic 3-fold X. Toda studied stability conditions, called the Gepner

type stability conditions, conjecturally corresponding to the Gepner point [100].

90

Chapter 3

Systolic inequality on K3 surfaces1

3.1 Introduction

Let (M, g) be a Riemannian manifold. Its systole sys(M, g) is defined to be the least

length of a non-contractible loop in M. In 1949, Charles Loewner proved that for any metric

g on the two-torus T2,

sys(T2, g)2 ≤ 2√3

vol(T2, g).

Moreover, the equality can be attained by the flat equilateral torus. There are various

generalizations of Loewner’s tours systolic inequality. We refer to [58] for a survey on the

rich subject of systolic geometry.

The first goal of the present chapter is to propose a new generalization of Loewner’s

torus systolic inequality from the perspective of Calabi–Yau geometry. We start with an

observation in the case of two-torus. Suppose that the torus is flat T2 ∼= C/Z + τZ, and is

equipped with the standard complex structure Ω = dz and symplectic structure ω = dx∧ dy.

Then the shortest non-contractible loops must be given by straight lines, which are nothing

but special Lagrangian submanifolds with respect to the complex and symplectic structures.

1Reference: [34].

91

Hence, under these assumptions, one can rewrite Loewner’s torus systolic inequality as:

infL:sLag

∣∣∣ ∫L

dz∣∣∣2 ≤ 1√

3

∣∣∣ ∫T2

dz ∧ dz∣∣∣.

Here “sLag" denotes the compact special Lagrangian submanifolds in T2 with respect to ω

and Ω = dz. The key observation is that the quantities in both sides of the above inequality

can be generalized to any Calabi–Yau manifold.

We propose the following definition of systole of a Calabi–Yau manifold, with respect to its

complex and symplectic structures.

Definition 3.1.1. Let Y be a Calabi–Yau manifold, equipping with a symplectic form ω and a

holomorphic top form Ω. Then its systole is defined to be

sys(Y, ω, Ω) := infL:sLag

∣∣∣ ∫L

Ω∣∣∣.

Here “sLag" denotes the compact special Lagrangian submanifolds in (Y, ω, Ω).

With this definition, we propose the following question that naturally generalizes

Loewner’s torus systolic inequality for Calabi–Yau manifolds.

Question 3.1.2. Let Y be a Calabi–Yau manifold and ω be a symplectic form on Y. Does there exist

a constant C > 0 such that

sys(Y, ω, Ω)2 ≤ C ·∣∣∣ ∫

YΩ ∧Ω

∣∣∣holds for any holomorphic top form Ω on Y?

Note that the ratio |∫

L Ω|2/|∫

Y Ω ∧Ω| has been considered in the context of attractor

mechanism in physics [31, 67, 72], which is of independent interest.

To answer Question 3.1.2, one needs to characterize the middle homology classes of

the Calabi–Yau manifold Y that can support special Lagrangian submanifolds. This is a

difficult problem in general. Nevertheless, if Y has a mirror Calabi–Yau manifold X, then

special Lagrangian submanifolds on Y conjecturally are corresponded to certain Bridgeland

semistable objects in the derived category of coherent sheaves on X [16]. The good news is

that when X is a K3 surface, there is a algebro-geometric result (Proposition 3.4.2) which

92

characterizes the classes that support Bridgeland semistable objects. This motivates us to

turn Question 3.1.2 into its mirror question (Question 3.1.4) and answer it in the case of K3

surfaces (Theorem 3.1.5). The formulation of the mirror question involves the categorical

analogues of systole and volume, which we now describe.

The second goal of the present chapter is to propose the definitions of categorical

systole and categorical volume, in terms of Bridgeland stability conditions on triangulated

categories. For the definition of categorical systole, there are two sources of motivation:

the correspondence between flat surfaces and stability conditions, and the conjectural

description of stability conditions on the Fukaya categories of Calabi–Yau manifolds.

To give a Bridgeland stability condition on a triangulated category D, one needs to

declare a set of objects in D to be semistable, and assign a complex number (central charge Z)

to each object in D. The correspondence between flat surfaces and stability conditions was

studied by Gaiotto–Moore–Neitzke [39], Bridgeland–Smith [17], and Haiden–Katzarkov–

Kontsevich [48]. Roughly speaking, under this correspondence, saddle connections on flat

surfaces are corresponded to semistable objects, and their lengths are corresponded to the

absolute values of central charges.

On the other hand, the systole of a flat surface is defined to be the length of its shortest

saddle connection. Based on the aforementioned correspondence, we propose the following

definition of systole of a Bridgeland stability condition.

Definition 3.1.3 (= Definition 3.2.1). Let D be a triangulated category, and σ be a Bridgeland

stability condition on D. Its systole is defined to be

sys(σ) := min|Zσ(E)| : E is σ−semistable.

This definition is supported by another source of motivation, the conjectural description

of stability conditions on the Fukaya categories of Calabi–Yau manifolds. Let Y be a Calabi–

Yau manifold and ω be a symplectic form on Y. One can associate a triangulated category,

the derived Fukaya category DπFuk(Y, ω), to the pair (Y, ω). Roughly speaking, the objects

in the Fukaya category are Lagrangian submanifolds with some extra data.

93

It is conjectured by Bridgeland [16] and Joyce [55] that a holomorphic top form Ω

on Y should give a Bridgeland stability condition on DπFuk(Y, ω), with central charges

given by period integrals of Ω along Lagrangians, and special Lagrangians are semistable

objects. Assuming this conjecture, Definition 3.1.1 coincides with Definition 3.1.3 for

D = DπFuk(Y, ω) and σ is given by a holomorphic top form Ω.

We summarize the correspondences in the following table.

Surface S Calabi–Yau (Y, ω) Triangulated categoryabelian differentials holomorphic top forms stability conditionssaddle connections special Lagrangians semistable objects

lengths period integrals central chargessys(S) sys(Y, ω, Ω) sys(σ)

|∫

Y Ω ∧Ω| vol(σ)

Table 3.1: Correspondence among flat surfaces, holomorphic top forms on Calabi–Yau manifolds, and stabilityconditions on triangulated categories.

In a previous joint work with Kanazawa and Yau [36], we study the categorical analogue

of the holomorphic volume∣∣∣ ∫Y Ω ∧Ω

∣∣∣ of Calabi–Yau manifolds. It is again motivated from

the conjectural description of stability conditions on their Fukaya categories. We recall the

definition of categorical volume vol(σ) in Section 3.2.

The third goal of the present chapter is to propose the mirror question of Question

3.1.2 under mirror symmetry, in terms of Bridgeland stability conditions. Mirror symmetry

conjecture states that Calabi–Yau manifolds come in pairs, in which the complex geometry of

one is equivalent to the symplectic geometry of the other, and vice versa. One mathematical

formulation of the conjecture, the homological mirror symmetry conjecture, was proposed by

Kontsevich [65] in 1994. It states that if (X, ωX, ΩX) and (Y, ωY, ΩY) is a mirror pair of

Calabi–Yau manifolds, then there are equivalences between triangulated categories

DbCoh(X, ΩX) ∼= DπFuk(Y, ωY) and DbCoh(Y, ΩY) ∼= DπFuk(X, ωX).

Under mirror symmetry, the complex moduli spaceMcpx(Y) should be identified with

the stringy Kähler moduli spaceMKah(X). The mathematical formulation of stringy Kähler

94

moduli space was proposed by Bridgeland [16]. The proposed definition is thatMKah(X)

is the double quotient Aut(D)\Stab∗(D)/C of certain subset Stab∗(D) ⊂ Stab(D) of the

space of stability conditions on the derived category of coherent sheaves D = DbCoh(X) on

X.

We can now formulate the mirror question of Question 3.1.2 in terms of Bridgeland

stability conditions.

Question 3.1.4. Let X be a Calabi–Yau manifold and D = DbCoh(X, Ω) be its derived category of

coherent sheaves. Does there exist a constant C > 0 such that

sys(σ)2 ≤ C · vol(σ)

holds for any σ ∈ Stab∗(D)?

Question 3.1.2 and Question 3.1.4 are mirror to each other in the sense that

• in Question 3.1.2, the symplectic form ω is fixed, and the question is asking for the

supremum of the ratio sys2/vol among all Ω ∈ Mcpx(Y); while

• in Question 3.1.4, the complex structure Ω is fixed, and the question is asking for the

supremum of the ratio sys2/vol among all [σ] ∈ Aut(D)\Stab∗(D)/C ∼= MKah(X).

Note that the ratio sys(σ)2/vol(σ) is invariant under the Aut(D)-action and the free

C-action by Lemma 3.2.3 and Lemma 3.2.8.

The subset Stab∗(D) ⊂ Stab(D) in Question 3.1.4 is well-defined for dim(X) ≤ 2. For X

an elliptic curve, Stab∗(D) = Stab(D) is the whole space of Bridgeland stability conditions.

For X a K3 surface, there is a precise definition of Stab∗(D) by Bayer and Bridgeland

[8] which we will recall in Section 3.4. However, when dim(X) ≥ 3, there is no general

characterization of the subset Stab∗(D) ⊂ Stab(D). We refer to [16, 36] for some attempts.

The fourth goal, which is the main result of the present chapter, is to give an affirmative

answer to Question 3.1.4 for K3 surfaces of Picard rank one.

95

Theorem 3.1.5 (= Theorem 3.4.1). Let X be a K3 surface of Picard rank one, with NS(X) = ZH

and H2 = 2n. Then

sys(σ)2 ≤ (n + 1)vol(σ)

holds for any σ ∈ Stab∗(DbCoh(X)).

We now sketch the proof of Theorem 3.1.5. The key algebro-geometric input is a result

of Bayer and Macrì [9, Theorem 6.8], which gives rise to a numerical criterion of classes that

support Bridgeland semistable objects (Proposition 3.4.2). This will show that the categorical

systole and categorical volume of a stability condition on D = DbCoh(X) only depend on

its central charge. Then one can use the description of Stab∗(D) in Bayer–Bridgeland [8] to

turn the systolic inequality into the following equivalent lattice-theoretic problem.

Question 3.1.6. Show that

min(s,d,r)∈Z3

sr≤nd2+1(s,d,r) 6=(0,0,0)

|s + 2n(β + iω)d + n(β + iω)2r|2 ≤ 4(n + 1)nω2

holds for any β ∈ R and ω > 0.

Note that (β, ω) in the inequality parametrizes the Bridgeland stability conditions lie in

Stab∗(D), and (s, d, r) parametrizes the classes that support Bridgeland semistable objects.

To answer Question 3.1.6, the most difficult part is to handle the case when ω is very close

to zero. This is done in Lemma 3.4.4. We then obtain the main result.

We also study the notion of spherical systoles of stability conditions. It is similar to

the notion of categorical systole except we only consider the masses of spherical objects

(Remark 3.4.5). The spherical systole and the categorical systole coincides for stability

conditions on the derived category of an elliptic curve. However, this is not true in general.

In particular, we show that the systolic inequality does not hold for spherical systoles of

stability conditions on the derived categories of K3 surface of Picard rank one (Proposition

3.4.6).

This chapter is organized as follows. In Section 3.2, we introduce the notions of

categorical systole and categorical volume. In Section 3.3, we give an affirmative answer to

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Question 3.1.4 for elliptic curves, which matches with Loewner’s torus systolic inequality

under mirror symmetry as expected. In Section 3.4, we give an affirmative answer to

Question 3.1.4 for K3 surfaces of Picard rank one by proving Theorem 3.1.5. In Section 3.5,

we indicate some directions for further studies.

3.2 Categorical systole and categorical volume

We refer to Section 2.2.1 for review on Bridgeland stability conditions.

3.2.1 Categorical systole

In the Introduction, we explained the two sources of motivation of our definition of

categorical systole: the connection between flat surfaces and stability conditions, and the

conjectural description of stability conditions on Fukaya categories of Calabi–Yau manifolds.

We recall the proposed definition of categorical systole, and investigate some basic

properties.

Definition 3.2.1. Let σ be a Bridgeland stability condition on D. Its systole is defined to be

sys(σ) := min|Zσ(E)| : E is σ−semistable.

Remark 3.2.2. The support property (Definition 2.2.1) of stability conditions guarantees that the

systole sys(σ) is a positive number and is attained by the absolute value of the central charge of a

stable object.

Below is an easy lemma.

Lemma 3.2.3. Let σ be a Bridgeland stability condition on D. Then

1. sys(σ) = minE:σ−stable |Zσ(E)| = min0 6=E∈D mσ(E). Here mσ(E) is the mass of E with

respect to the stability condition σ (Definition 2.2.1).

2. sys(Φ · σ) = sys(σ) for any autoequivalence Φ ∈ Aut(D).

3. sys(z · σ) = ex · sys(σ) for any complex number z = x + iy ∈ C.

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Proof. The first statement follows directly from the definition of mass. The second and third

statements follow from the definition of the actions of Aut(D) and C.

Remark 3.2.4. By Lemma 3.2.3 (2), the categorical systole defines a real-valued function on the

quotient space Stab(D)/Aut(D). It is shown in Bridgeland–Smith [17] that this quotient space is

isomorphic to the space of certain meromorphic quadratic differentials with simple zeros on a marked

Riemann surface, provided D is the CY3 triangulated category associated to the surface. Under

this correspondence, the categorical systole is related to the period integrals∫ √

φ along the saddle

connections of the quadratic differential φ.

Proposition 3.2.5. Categorical systole defines a continuous function on the space of Bridgeland

stability conditions:

Stab(D) −→ R>0, σ 7→ sys(σ).

Proof. Let σ1, σ2 ∈ Stab(D) be two Bridgeland stability conditions with d(σ1, σ2) < ε. By

Remark 3.2.2, sys(σ1) = mσ1(E) for some object E ∈ D. Thus

log sys(σ1) = log mσ1(E) > log mσ2(E)− ε ≥ log sys(σ2)− ε.

Similarly, we have log sys(σ2) > log sys(σ1)− ε. Hence log sys(σ) is a continuous function

on Stab(D), and so is sys(σ).

Example 3.2.6 (Derived category of A2-quiver). Let Q be the A2-quiver (· → ·) and D =

Db(Rep(Q)) be the bounded derived category of the category of representations of Q. Let E1 and E2

be the simple objects in Rep(Q) with dimension vectors dim(E1) = (1, 0) and dim(E2) = (0, 1).

There is one more indecomposable object E3 that fits into the exact sequence

0→ E2 → E3 → E1 → 0.

Let σ = (Z ,P) be a stability condition on D with z1 := Z(E1) and z2 := Z(E2). Suppose that

P(0, 1] = Rep(Q). Then

• When arg(z1) < arg(z2), the only σ-stable objects are E1 and E2 up to shiftings. Thus

sys(σ) = min|z1|, |z2|.

98

• When arg(z1) > arg(z2), the only σ-stable objects are E1, E2 and E3 up to shiftings. Thus

sys(σ) = min|z1|, |z2|, |z1 + z2|.

Note that in order to compute the categorical systoles of stability conditions in different chambers,

one needs to compute the central charges of different sets of dimensional vectors. However, it is not

the case for the derived categories of elliptic curves and K3 surfaces, see the proof of Theorem 3.3.1

and Proposition 3.4.2.

3.2.2 Categorical volume

Motivated by the discussion in Section 2.3.2, we define the notion of categorical volume of

Bridgeland stability conditions. It is the categorical analogue of the holomorphic volume∣∣∣ ∫Y Ω ∧Ω∣∣∣ of a compact Calabi–Yau manifold Y with holomorphic top form Ω.

Definition 3.2.7. Let Ei be a basis of the numerical Grothendieck groupN (D) and let σ = (Z ,P)

be a Bridgeland stability condition on D. Its volume is defined to be

vol(σ) :=∣∣∣∑

i,jχi,jZ(Ei)Z(Ej)

∣∣∣,where (χi,j) = (χ(Ei, Ej))

−1 is the inverse matrix of the Euler pairings.

One can easily check that the above definition is independent of the choice of the basis

Ei. The following lemma is also straightforward.

Lemma 3.2.8. Let σ be a Bridgeland stability condition on D. Then

1. vol(Φ · σ) = vol(σ) for any autoequivalence Φ ∈ Aut(D).

2. vol(z · σ) = e2x · vol(σ) for any complex number z = x + iy ∈ C.

Remark 3.2.9. Although the categorical volume can be defined for stability conditions on any

triangulated category D, its geometric meaning is not clear unless D comes from compact Calabi–Yau

geometry. Below is an example of Calabi–Yau triangulated category for which the categorical volume

vanishes for some stability conditions.

99

Example 3.2.10 (CY3-category of A2-quiver). Let Q be the A2-quiver (· → ·) and ΓQ be the

Ginzburg Calabi–Yau–3 dg-algebra associated with Q. See [41, 60] for the definition of ΓQ. Let

D(ΓQ) be the derived category of dg-modules over ΓQ and D = Dfd(ΓQ) be the full subcategory of

D(ΓQ) consisting of dg-modules with finite dimensional total cohomology.

By Keller [60, Theorem 6.3], the category D is a Calabi–Yau–3 category, i.e., there is a natural

isomorphism Hom(E, F) ∼= Hom(F, E[3])∗ for any E, F ∈ D. By Smith [91], there is an embedding

of D into the Fukaya category of certain quasi-projective Calabi–Yau threefold.

The numerical Grothendieck group N (D) is generated by two spherical objects S1, S2, each

associated with a vertex in the A2-quiver. The spherical objects satisfy:

• Hom∗D(S1, S1) = Hom∗D(S2, S2) = C⊕C[−3].

• Hom∗D(S1, S2) = C[−1], Hom∗D(S2, S1) = C[−2].

Let σ = (Z ,P) be a stability condition on D with z1 := Z(S1) and z2 := Z(S2). Then its

categorical volume is

vol(σ) = |z1z2 − z2z1| = 2|Im(z1z2)|,

which vanishes if z1z2 ∈ R. This can happen if z1 and z2 are of the same phase, i.e., the stability

condition σ sits on a wall in Stab(D).

3.3 Elliptic curves case

In this section, we give an affirmative answer to Question 3.1.4 in the case of elliptic

curves. Let D = DbCoh(E) be the derived category of an elliptic curve E. In this case,

the ˜GL+(2; R)-action on the space of Bridgeland stability conditions Stab(D) is free and

transitive [14, Theorem 9.1]. Hence,

Stab(D) ∼= ˜GL+(2; R) ∼= C×H,

and the double quotient

Aut(D)\Stab(D)/C ∼= PSL(2, Z)\H

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is indeed the Kähler moduli space of elliptic curve. Thus we should take Stab∗(D) = Stab(D)

in Question 3.1.4 to be the whole space of stability conditions.

Theorem 3.3.1. Let D = DbCoh(E) be the derived category of an elliptic curve E. Then

sys(σ)2 ≤ 1√3· vol(σ)

holds for any σ ∈ Stab(D).

One can consider this inequality as the mirror of Loewner’s torus systolic inequality in

the introduction.

Proof. Firstly, by Lemma 3.2.3 and Lemma 3.2.8, one notices that the ratio

sys(σ)2

vol(σ)

is invariant under the free C-action on the space of stability conditions. Hence we only need

to compute the ratio on the quotient space Stab(D)/C ∼= H.

The quotient space Stab(D)/C ∼= H can be parametrized by the normalized stability

conditions as follows. Let τ = β + iω ∈ H where β ∈ R and ω > 0. The associated

normalized stability condition στ is given by:

• Central charge: Zτ(F) = −deg(F) + τ · rk(F).

• For 0 < φ ≤ 1, the set of (semi)stable objects Pτ(φ) consists of the slope-(semi)stable

coherent sheaves whose central charge lies in the ray R>0 · eiπφ.

• For other φ ∈ R, define Pτ(φ) by the property Pτ(φ + 1) = Pτ(φ)[1].

Note that there is no wall-crossing phenomenon in the elliptic curve case, i.e., for any

Bridgeland stability condition on DbCoh(E), the set of all Bridgeland (semi)stable objects is

the same as the set of all slope-(semi)stable coherent sheaves up to shiftings. This makes the

computation of categorical systole easier.

To compute the systole of στ, by Lemma 3.2.3 (1), we need to know the central charges

of all the stable objects of στ, which are in this case the slope-stable coherent sheaves. Recall

101

that if F is a slope-stable coherent sheaf on E, then it is either a vector bundle or a torsion

sheaf. The slope-stable vector bundles on an elliptic curve E are well-understood, see for

instance [5, 81].

• Let F be an indecomposable vector bundle of rank r and degree d on an elliptic curve

E. Then F is slope-stable if and only if d and r are relatively prime.

• Fix a point x ∈ E. For every rational number µ = dr , where r > 0 and (d, r) = 1, there

exists a unique slope-stable vector bundle Vµ of rank r and det(Vµ) ∼= OE(dx).

Hence the categorical systole is

sys(στ) = min(d,r)=1

r>0

1, | − d + τr| = min(d,r)∈Z2

(d,r) 6=(0,0)

| − d + τr| = λ1(Lτ),

where λ1(Lτ) denotes the least length of a nonzero element in the lattice Lτ = 〈1, τ〉.

On the other hand, the categorical volume of στ has been computed in Example 2.3.5,

which equals to 2ω. Thus

supτ∈H

sys(στ)2

vol(στ)=

12· sup

τ∈H

λ1(Lλ)2

ω.

Note that ω is the area of the parallelogram spanned by 1 and τ. Hence the quantity

supτ∈H λ1(Lλ)2/ω is the so-called Hermite constant γ2 of lattices in R2. It is classically known

that the Hermite constant γ2 = 2√3

(see for instance [19]). This concludes the proof.

3.4 K3 surfaces case

In this section, we give an affirmative answer to Question 3.1.4 for K3 surfaces of Picard

rank one. This is the main result of the present chapter. We start with recalling some

standard notations.

Let X be a smooth complex projective K3 surface and D = DbCoh(X) be its derived

category. Sending an object E ∈ D to its Mukai vector v(E) = ch(E)√

td(X) identifies the

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numerical Grothendieck group of D with the lattice

N (D) = H0(X; Z)⊕NS(X)⊕ H4(X; Z).

The Mukai pairing on N (D) is given by

((r1, D1, s1), (r2, D2, s2)) = D1 · D2 − r1s2 − r2s1.

Note that the Mukai pairing on N (D) is non-degenerate, hence any group homomorphism

Z : N (D)→ C can be written as

Z(v) = (Ω, v) : N (D)→ C

for some unique Ω ∈ N (D)⊗C.

Define the period domains

Q±(X) = Ω ∈ N (D)⊗C : (Ω, Ω) = 0, (Ω, Ω) > 0,

Q±0 (X) = Q±(X)\⋃

δ∈∆(X)

δ⊥.

Here ∆(X) = δ ∈ N (D) : δ2 = −2 denotes the root system. We let Q+(X) be the con-

nected component containing classes (1, iω,− 12 ω2) for ω ∈ NS(X) ample. These domains

are invariant under the rescaling action of C∗ on N (D)⊗C.

Define the tube domain

T ±(X) = β + iω : β, ω ∈ NS(X)⊗R, (ω, ω) > 0,

where T +(X) denotes the connected component containing iω for ample classes ω. One

can check that the map

T +(X) → Q+(X), β + iω 7→ Ω = exp(β + iω)

is an embedding which gives a section of the C∗-action on Q+(X).

We now briefly review some previous results on Bridgeland stability conditions on the

derived categories D = DbCoh(X) of K3 surfaces. The existence of stability conditions on

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D was first proved by Bridgeland [15]. The main result of [15] is a complete description of

the distinguished connected component Stab†(D) ⊂ Stab(D) containing the set of geometric

stability conditions, for which all skyscraper sheaves Ox are stable and of the same phase.

Let Stab∆(D) ⊂ Stab(D) be the union of those connected components which are images

of Stab†(D) under the Aut(D)-actions. The complex submanifold Stab∗(D) ⊂ Stab∆(D) is

defined to be the subset consisting of stability conditions σ = (Z = (Ω,−),P) such that

the corresponding vector Ω ∈ N (D)⊗C satisfies (Ω, Ω) = 0. Note that this submanifold is

denoted by Stab∗red(X) in [8].

In the work of Bayer and Bridgeland [8], they define the stringy Kähler moduli space of the

K3 surface X as the quotient

MKah(X) := (Stab∗(D)/C)/(AutCY(D)/[2]),

where AutCY(D) ⊂ Aut(D) is a certain subgroup of Aut(D) consisting of Calabi–Yau

autoequivalences. The mirror symmetry phenomenon in this context is that the complex

orbifoldMKah(X) can be identified with the base of Dolgachev’s family of lattice-polarized

K3 surfaces mirror to X [30]. We refer to [8] for more details.

We now state the main theorem.

Theorem 3.4.1. Let X be a K3 surface of Picard rank one, with NS(X) = ZH and H2 = 2n. Then

sys(σ)2 ≤ (n + 1)vol(σ)

holds for any σ ∈ Stab∗(DbCoh(X)).

The following proposition allows us to compute the categorical systole of a stability

condition σ = (Zσ,Pσ) ∈ Stab∆(D) using only its central charge Zσ.

Proposition 3.4.2. Let σ = (Zσ,Pσ) ∈ Stab∆(D). Then

sys(σ) = min|Zσ(v)| : 0 6= v ∈ N (D), v2 = (v, v) ≥ −2.

Proof. It suffices to prove the statement for σ ∈ Stab†(D), since autoequivalences induce

Hodge isometries on the numerical Grothendieck group N (D), and Stab∆(D) is the image

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of the connected component Stab†(D) under the Aut(D)-actions.

Let v = mv0 ∈ N (D) be a Mukai vector, where m ∈ Z>0 and v0 is primitive. A result of

Bayer and Macrì [9, Theorem 6.8], which is based on a previous result of Toda [98], says that

if v20 ≥ −2, then there exists a σ-semistable object with Mukai vector v for any σ ∈ Stab†(D).

Hence

sys(σ) = min|Zσ(E)| : E is σ−semistable

≤ min|Zσ(v)| : 0 6= v ∈ N (D), v2 ≥ −2.

On the other hand, for any stable object E,

v(E)2 = −χ(E, E) = − hom0(E, E) + hom1(E, E)− hom2(E, E)

= −2 + hom1(E, E) ≥ −2.

Hence

sys(σ) = min|Zσ(E)| : E is σ−stable

≥ min|Zσ(v)| : 0 6= v ∈ N (D), v2 ≥ −2.

This concludes the proof.

Remark 3.4.3. By Proposition 3.4.2, the set of Mukai vectors needed for computing the categorical

systole is independent of stability condition σ ∈ Stab∆(D). This is also the case for elliptic curves,

where the categorical systole is the minimum among the absolute values of central charges of all the

nonzero vectors (d, r) ∈ Z2. However, this is not true in general, see for instance Example 3.2.6.

We now prove the main theorem.

Proof of Theorem 3.4.1. Let X be a K3 surface of Picard rank one, with NS(X) = ZH and

H2 = 2n. The numerical Grothendieck group of D = DbCoh(X) is

N (D) = Z⊕ZH ⊕Z.

By Proposition 3.4.2 and the definition of categorical volume, in order to compute the

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categorical systoles and volumes of stability conditions σ ∈ Stab∗(D), it suffices to know

their central charges.

By [8, 15], up to autoequivalences (which do not change the ratio sys(σ)2/vol(σ)),

Z = (Ω,−) is the central charge of a stability condition in the submanifold Stab∗(D) if and

only if Ω ∈ Q+0 (X). Recall that Q+

0 (X) ⊂ Q+(X) admits a section of the C∗-action given

by the tube domain. Hence Ω can be written as Ω = c exp((β + iω)H) for some nonzero

complex number c, and β ∈ R, ω > 0.

By Proposition 3.4.2 and 2.4.6,

supσ∈Stab∗(D)

sys(σ)2

vol(σ)= sup

Ω=exp((β+iω)H)∈Q+0 (X)

min|(Ω, v)|2 : v2 ≥ −2, v 6= 04nω2

= supΩ=exp((β+iω)H)∈Q+(X)

min|(Ω, v)|2 : v2 ≥ −2, v 6= 04nω2

The second equality follows from the fact that the numerator vanishes if Ω ∈ δ⊥.

Let v = (r, dH, s) ∈ Z⊕ZH ⊕Z. One then can plug in the integers r, d, s and the real

numbers β, ω to the central charge (Ω, v), and rewrite the above quantity as

supβ∈Rω>0

min|s + 2n(β + iω)d + n(β + iω)2r|2 : (s, d, r) ∈ Z3\(0, 0, 0), sr ≤ nd2 + 14nω2 .

In order to deal with the situation when ω is very close to zero, we need the following

technical lemma.

Lemma 3.4.4. For any real number β and any 0 < ω < 1√n , there exists integers (s, d, r) such that:

1 ≤ r ≤ 1√nω

,

|s + 2nβd + nβ2r| <√

nω,

0 ≤ nd2 − sr ≤ n.

Proof of Lemma 3.4.4. Let l = b 1√nωc+ 1. For each 1 ≤ j ≤ l, choose dj ∈ Z such that

−12< dj + jβ ≤ 1

2.

106

Consider the real numbers 2nβdj + nβ2 j1≤j≤l modulo 1. There is at least a pair

(2nβdj + nβ2 j, 2nβdk + nβ2k) has distance less than or equals to 1/l modulo 1. Say j > k

without loss of generality. We choose r = j− k, d = dj − dk, and choose s to be the integer

closest to −2nβd− nβ2r. Then

1 ≤ r ≤ b 1√nωc

and

|s + 2nβd + nβ2r| ≤ 1b 1√

nωc+ 1

<√

nω.

Let ε = s + 2nβd + nβ2r. Then

nd2 − sr = nd2 − (−2nβd− nβ2r + ε)r

= n(d + rβ)2 − rε.

We have

(d + rβ)2 = ((dj + jβ)− (dk + kβ))2 < 1

and

|rε| < 1√nω·√

nω = 1.

Hence −1 < nd2 − sr < n + 1. Since it is an integer, thus 0 ≤ nd2 − sr ≤ n.

We are now ready to finish the proof Theorem 3.4.1. By previous discussion, it suffices

to prove the following claim.

Claim. For any β ∈ R and ω > 0, there exists (s, d, r) ∈ Z3 not all zero such that sr ≤ nd2 + 1

and|s + 2n(β + iω)d + n(β + iω)2r|2

4nω2 < n + 1.

• If ω ≥ 1√n , one can simply take the class of skyscraper sheaves (s, d, r) = (1, 0, 0) and

check that it satisfies the two inequalities above.

107

• If ω < 1√n , we choose (s, d, r) as in Lemma 3.4.4. Then it satisfies sr < nd2 + 1 and

|s + 2n(β + iω)d + n(β + iω)2r|24nω2 =

14n

( s + 2nβd + nβ2rω

+ nωr)2

+ (nd2 − rs)

<1

4n(√

n +√

n)2 + n = n + 1.

This concludes the proof of Theorem 3.4.1.

Remark 3.4.5 (Spherical systole). The notion of spherical objects in a triangulated category was

introduced by Seidel and Thomas [89]. These objects are the categorical analogue of Lagrangian

spheres in derived Fukaya categories. An object S ∈ DbCoh(X) in the derived category of coherent

sheaves on a Calabi–Yau n-fold X is called spherical if Hom∗D(S, S) = C⊕C[−n].

One can define the spherical systole of a Bridgeland stability condition σ on a triangulated

category D as the minimum among masses of spherical objects:

syssph(σ) := minmσ(S) : S ∈ D spherical.

Let D = DbCoh(E) be the derived category of coherent sheaves on an elliptic curve E. Then the

categorical systole of any stability condition on D is achieved by some spherical object in D, hence

sys(σ) = syssph(σ) for any σ ∈ Stab(D). However, this is not true for the derived categories of K3

surfaces.

The following proposition shows that the systolic inequality does not hold for spherical

systole on K3 surfaces.

Proposition 3.4.6. Let X be a K3 surface of Picard rank one and D = DbCoh(X) be its derived

category of coherent sheaves. Then

supσ∈Stab∗(D)

syssph(σ)2

vol(σ)= +∞.

Proof. Let NS(X) = ZH and H2 = 2n. Let ωH ∈ NS(X) ⊗R be an ample class. Then

Z = (exp(iωH),−) gives the central charge of a stability condition in Stab∗(D) if ω > 1

108

([15, Lemma 6.2]). By Proposition 3.4.2 and 2.4.6,

supσ∈Stab∗(D)

syssph(σ)2

vol(σ)≥ sup

ω>1

min|(exp(iωH), v)|2 : v2 = −24nω2 .

Here we use the fact that the Mukai vector of a spherical object S satisfies v(S)2 =

− hom0(S, S) + hom1(S, S)− hom2(S, S) = −2.

Let v = (r, dH, s) ∈ Z⊕NS(X)⊕Z be a vector satisfying v2 = 2nd2 − 2rs = −2. Then

rs 6= 0 and r, s are both positive or both negative. Hence

supσ∈Stab∗(D)

syssph(σ)2

vol(σ)≥ sup

ω>1

min|(exp(iωH), v)|2 : v2 = −24nω2

= supω>1

minnd2−rs=−1

14n

( sω

+ nωr)2− 1

≥ supω>1

14n

( 1ω

+ nω)2− 1 = +∞.

Remark 3.4.7. In the proof of Theorem 3.4.1, we do not make use of the Mukai vectors of spherical

objects. Recall that

sys(σ) = min|Zσ(v)| : v2 ≥ −2, v 6= 0,

but we only use the Mukai vectors that satisfy 0 ≤ v2 ≤ 2n to prove the systolic inequality

(c.f. Lemma 3.4.4).

3.5 Further studies

3.5.1 Optimal systolic ratio of K3 surfaces of Picard rank one

It is not clear whether the constant n + 1 in the categorical systolic inequality

sys(σ)2 ≤ (n + 1)vol(σ)

109

is optimal. As discussed in the proof of Theorem 3.4.1, finding the optimal systolic ratio is

equivalent to solving the following lattice-theoretic problem:

supβ∈Rω>0

min(s,d,r)∈Z3

sr≤nd2+1(s,d,r) 6=(0,0,0)

|s + 2n(β + iω)d + n(β + iω)2r|24nω2 =?

One might expect the optimal is achieved by certain stability conditions on K3 surfaces

with “extra symmetries", like the optimal of Loewner’s torus systolic inequality is achieved

by the flat equilateral torus.

3.5.2 Systolic inequality for K3 surfaces of higher Picard rank

Almost the whole argument of the proof of Theorem 3.4.1 can be applied to K3 surfaces

of higher Picard rank. In particular, Bayer–Bridgeland’s description of stringy Kähler

moduli space, Proposition 3.4.2 and 2.4.6 all work for general K3 surfaces. Hence the systolic

problem is equivalent to the following:

supβ,ω∈NS(X)⊗R

ω2>0

min(s,D,r)∈Z⊕NS(X)⊕Z

2sr≤D2+2(s,D,r) 6=(0,0,0)

|s + (β + iω).D + 12 (β + iω)2r|2

2ω2 =?

The problem becomes more complicated for K3 surfaces of higher Picard rank because

of the indefiniteness of intersection pairing on NS(X).

3.5.3 Miscellaneous

• It is mentioned in Remark 3.4.3 that in the case of elliptic curves and K3 surfaces, the

set of Mukai vectors needed for computing the categorical systole is independent of

stability conditions. This is also true for the derived categories of coherent sheaves on

smooth projective curves of genus ≥ 1, since there is no wall-crossings on the space of

stability conditions [70]. It would be interesting to find other triangulated categories

that also satisfy this property.

• Akrout [3] proves that the systole of Riemann surfaces is a topological Morse function on

110

the Teichmüller space. It would be interesting to know whether the categorical systole

is also a topological Morse function on the space of Bridgeland stability conditions.

111

Chapter 4

Entropy of autoequivalences1

4.1 Introduction

In the pioneering work [29], Dimitrov–Haiden–Katzarkov–Kontsevich introduced the

notion of categorical entropy of an endofunctor on a triangulated category with a split

generator. A typical example of such triangulated category is given by the bounded

derived category Db(X) of coherent sheaves on a variety X. When X is a smooth projective

variety with ample (anti-)canonical bundle, the group of autoequivalences on Db(X) is

well-understood [12]. It is generated by tensoring line bundles, automorphisms on the

variety, and degree shifts. The categorical entropy in this case was computed by Kikuta and

Takahashi [63]. On the other hand, when the variety is Calabi–Yau, there are much more

autoequivalences on Db(X) because of the presence of spherical objects.

In this article, we show that the composite of the simplest spherical twist TOX with

−⊗O(−H) already gives an interesting categorical entropy.

Theorem 4.1.1 (= Theorem 4.3.1). Let X be a strict Calabi–Yau manifold over C of dimension

d ≥ 3. Consider the autoequivalence Φ := TOX (−⊗O(−H)) on Db(X). The categorical entropy

1References: [32, 33].

112

ht(Φ) is a positive function in t ∈ R. Moreover, for any t ∈ R, ht(Φ) is the unique λ > 0 satisfying

∑k≥1

χ(O(kH))

ekλ= e(d−1)t.

A simple argument on Hilbert polynomial shows that this equation defines an algebraic

curve over Q in the coordinate (et, eλ). Thus the algebraicity conjecture in [29, Question 4.1]

holds in this case.

The notion of categorical entropy is a categorical analogue of topological entropy of a

continuous surjective self-map on a compact metric space. There is a fundamental theorem

of topological entropy due to Gromov and Yomdin.

Theorem 4.1.2 ([44, 45, 106]). Let M be a compact Kähler manifold and let f : M → M be a

surjective holomorphic map. Then

htop( f ) = log ρ( f ∗),

where htop( f ) is the topological entropy of f , and ρ( f ∗) is the spectral radius of f ∗ : H∗(M; C)→

H∗(M; C).

Kikuta and Takahashi proposed the following analogous conjecture on categorical

entropy.

Conjecture 4.1.3 ([63]). Let X be a smooth proper variety over C and let Φ be an autoequivalence

on Db(X). Then

h0(Φ) = log ρ(HH•(Φ)),

where HH•(Φ) : HH•(X) → HH•(X) is the induced C-linear isomorphism on the Hochschild

homology group of X, and ρ(HH•(Φ)) is its spectral radius.

Note that one can replace HH•(Φ) by the induced Fourier-Mukai type action on the

cohomology ΦH∗ : H∗(X; C)→ H∗(X; C), because there is a commutative diagram

HH•(X)HH•(Φ) //

IXK

HH•(X)

IXK

H∗(X; C)

ΦH∗ // H∗(X; C),

113

where IXK is the modified Hochschild–Kostant–Rosenberg isomorphism [71, Theorem 1.2].

Hence ρ(HH•(Φ)) = ρ(ΦH∗).

Conjecture 4.1.3 has been proved in several cases. See [61, 62, 63, 107].

Now we explain the motivation of the present work, namely why we should not expect

Conjecture 4.1.3 to hold in general. We first note that Theorem 4.1.2 does not hold if f is

not holomorphic. For example, there is a construction by Thurston [95] of pseudo-Anosov

maps that act trivially on the cohomology, and thus have zero spectral radius. Moreover,

Dimitrov–Haiden–Katzarkov–Kontsevich [29, Theorem 2.18] showed that the categorical

entropy of the induced autoequivalence on the derived Fukaya category is equal to log λ,

where λ > 1 is the stretch factor of the pseudo-Anosov map. Hence the analogous statement

of Conjecture 4.1.3 is not true if Db(X) is replaced by the derived Fukaya categories of the

symplectic manifolds considered in [29, 95].

Motivated by homological mirror symmetry, one may expect to find counterexamples of

Conjecture 4.1.3 on the derived categories of coherent sheaves on Calabi–Yau manifolds. In

other words, the discrepancy between complex and symplectic geometry should lead to the

discrepancy between Theorem 4.1.2 and Conjecture 4.1.3.

Using Theorem 4.1.1, we construct counterexamples of Conjecture 4.1.3.

Proposition 4.1.4 (= Proposition 4.4.1). For any even integer d ≥ 4, let X be a Calabi–Yau

hypersurface in CPd+1 of degree (d + 2) and Φ = TOX (−⊗O(−1)). Then

h0(Φ) > 0 = log ρ(ΦH∗).

In particular, Conjecture 4.1.3 fails in this case.

Interestingly, as pointed out to the author by Genki Ouchi, the same autoequivalence

does not produce counterexamples of Conjecture 4.1.3 if X is an odd dimensional Calabi–Yau

manifold (see Remark 4.4.3).

114

4.2 Preliminaries

4.2.1 Categorical entropy

We recall the notion of categorical entropy introduced by Dimitrov-Haiden-Katzarkov-

Kontsevich [29].

Let D be a triangulated category. A triangulated subcategory is called thick if it is closed

under taking direct summands. The split closure of an object E ∈ D is the smallest thick

triangulated subcategory containing E. An object G ∈ D is called a split generator if its split

closure is D.

Definition 4.2.1 ([29]). Let E and F be non-zero objects in D. If F is in the split closure of E, then

the complexity of F relative to E is defined to be the function

δt(E, F) := inf

k

∑i=1

enit

∣∣∣∣∣∣0 A1 . . . Ak−1 F⊕ F′

E[n1] . . . E[nk]

//

__ //

__ ,

where t is a real parameter that keeps track of the shiftings.

Note that when the category D is Z2-graded in the sense that [2] ∼= idD , only the value

at t equals to zero, δ0(E, F), will be of any use.

Definition 4.2.2 ([29]). Let D be a triangulated category with a split generator G and let Φ : D →

D be an endofunctor. The categorical entropy of Φ is defined to be the function ht(Φ) : R →

[−∞, ∞) given by

ht(Φ) := limn→∞

1n

log δt(G, ΦnG).

Lemma 4.2.3 ([29]). The limit limn→∞1n log δt(G, ΦnG) exists in [−∞, ∞) for every t ∈ R, and

is independent of the choice of the split generator G.

We will use the following proposition to compute categorical entropy.

Proposition 4.2.4 ([29, 63]). Let G and G′ be split generators of Db(X) and let Φ be an autoequiv-

alence on Db(X). Then the categorical entropy equals to

ht(Φ) = limn→∞

1n

log ∑l∈Z

dim HomlDb(X)(G, ΦnG′)e−lt.

115

4.2.2 Spherical objects and spherical twists

We recall the notions of spherical objects and spherical twists introduced by Seidel-

Thomas [89]. They are the categorical analogue of Lagrangian spheres in symplectic

manifolds and the Dehn twists along Lagrangian spheres.

Definition 4.2.5 ([89]). An object S ∈ Db(X) is called spherical if S⊗ωX ∼= S and Hom•Db(X)(S, S) =

C⊕C[−dim X].

Definition 4.2.6 ([89]). The spherical twist TS with respect to a spherical object S is an autoequiv-

alence on Db(X) given by

E 7→ TS(E) := Cone(Hom•Db(X)(S, E)⊗ S→ E).

We also recall the definition of strict Calabi–Yau manifolds.

Definition 4.2.7. A smooth projective variety X is called strict Calabi–Yau if ωX ∼= OX and

Hi(X,OX) = 0 for all 0 < i < dim X. This is equivalent to the condition that OX is a spherical

object.

4.3 Computation of categorical entropy

We fix the notations and assumptions that will be used throughout this section. We work

over complex number field C.

Notations

• X is a strict Calabi–Yau manifold (Definition 4.2.7) with a very ample line bundle H.

• d := dimCX ≥ 3.

• O := OX and O(k) := OX(kH).

• ak := h0(O(k)) = χ(O(k)) for k > 0.

116

• G := ⊕d+1i=1O(i) and G′ := ⊕d+1

i=1O(−i). By a result of Orlov [79], both G and G′ are

split generators of Db(X).

The goal of this section is to prove the following theorem.

Theorem 4.3.1. Let X be a strict Calabi–Yau manifold over C of dimension d ≥ 3. Consider the

autoequivalence Φ := TO (−⊗O(−1)) on Db(X). The categorical entropy ht(Φ) is a positive

function in t ∈ R. Moreover, for any t ∈ R, ht(Φ) is the unique λ > 0 satisfying

∑k≥1

χ(O(k))ekλ

= e(d−1)t.

We begin with a lemma that will be crucial in the computation of categorical entropy.

Lemma 4.3.2. 1. For any integers n ≥ 0 and k > 0, Homl(O, Φn(G′) ⊗ O(−k)) is zero

except for l = d, d + (d− 1), . . . , d + n(d− 1).

2. For n ≥ 0 and k > 0, define

Bn,k :=n

∑m=0

dim Homd+m(d−1)(O, Φn(G′)⊗O(−k)) · e−m(d−1)t.

In particular, B0,k = dim Homd(O,O(−k)) = ak. There is a recursive relation among Bn,k’s:

Bn,k = Bn−1,k+1 + ake−(d−1)tBn−1,1. (*)

Proof. We prove the first assertion by induction on n. The statement is true when n = 0 by

Kodaira vanishing theorem and Serre duality.

By the definition of Φ, there is an exact triangle:

Φn−1(G′)⊗O(−1)→ Φn(G′)→ Hom•(O, Φn−1(G′)⊗O(−1))⊗O[1] +1−→ .

By tensoring it with O(−k) and applying Hom•(O,−), we get a long exact sequence:

Hom•(O, Φn−1(G′)⊗O(−k− 1))→ Hom•(O, Φn(G′)⊗O(−k))

→ Hom•(O, Φn−1(G′)⊗O(−1))⊗Hom•(O,O(−k))[1] +1−→ · · ·

117

Suppose the statement is true for n− 1. Then the first complex in the long exact sequence

is non-zero only at degree d, d + (d− 1), . . . , d + (n− 1)(d− 1), and the third complex is

non-zero only at degree d + (d− 1), d + 2(d− 1), . . . , d + n(d− 1). Since we assume the

dimension d ≥ 3, the long exact sequence splits into short exact sequences, and the proof

follows.

The recursive relation in the second assertion also follows from the above long exact

sequence.

Lemma 4.3.3. Define

Ps,k := ∑i1+···+iq=s

i1≥k

ai1 ai2 · · · aiq e−q(d−1)t,

where the summation runs over all ordered partitions of s with the first piece no less than k. Then we

have

Bn,k = an+k +n

∑s=1

asPn+k−s,k.

Proof. Induction on n and use the recursive relation (*).

Now we are ready to prove Theorem 4.3.1.

Proof of Theorem 4.3.1. By Proposition 4.2.4, the categorical entropy can be written as

ht(Φ) = limn→∞

1n

log ∑l

dim Homl(G, Φn(G′))e−lt

= limn→∞

1n

logd+1

∑k=1

∑l

dim Homl(O, Φn(G′)⊗O(−k))e−lt

= limn→∞

1n

logd+1

∑k=1

Bn,k.

By Lemma 4.3.3 and the fact that ak is an increasing sequence, we have Bn,1 ≤ Bn,k+1 ≤

Bn+k,1. Hence

ht(Φ) = limn→∞

1n

log Bn,1.

118

Define Cn := Bn,1e−(d−1)t and a′k := ake−(d−1)t. Then again by Lemma 4.3.3,

Cn =(

an+1 +n

∑s=1

asPn+1−s,1

)e−(d−1)t

= ∑i1+···+iq=n+1

a′i1 a′i2 · · · a′iq

.

Thus

Cn = a′1Cn−1 + a′2Cn−2 · · ·+ a′nC0 + a′n+1.

Notice that ht(Φ) = limn→∞1n log Bn,1 = limn→∞

1n log Cn is a positive real number

for any t ∈ R, because there exists some k > 0 (depending on d and t) such that a′k =

ake−(d−1)t > 1, hence

limn→∞

1n

log Cn ≥ limn→∞

1n

log(a′k)b n+1

k c =1k

log a′k > 0.

We claim that the categorical entropy ht(Φ) is the unique λ > 0 satisfying

∑k≥1

a′kekλ

= 1.

Or equivalently,

∑k≥1

χ(O(k))ekλ

= e(d−1)t.

This will conclude the proof of the theorem. We prove the claim in the following lemma,

with a slight change in notations.

Lemma 4.3.4. Let ann≥1 be a sequence of positive real numbers, and define Cnn≥0 inductively

by

Cn = a1Cn−1 + a2Cn−2 · · ·+ anC0 + an+1. (**)

Suppose that

limn→∞

1n

log Cn = λ > 0,

then we have

∑k≥1

ak

ekλ= 1.

119

Proof. Assume that ∑k≥1akekλ > 1. Then there exists some m > 0 so that ∑m

k=1akekλ > 1.

Moreover, there exists some λ1 > λ such that ∑mk=1

akekλ > ∑m

k=1ak

ekλ1> 1.

Choose a constant D1 > 0 such that Ck > D1e(k+1)λ1 for all 0 ≤ k ≤ m. We can then

prove by induction that Cn > D1e(n+1)λ1 holds for all n ≥ 0. Indeed, by the induction

hypothesis, (**) and ∑mk=1

akekλ1

> 1, for n > m, we have

Cn > a1Cn−1 + · · ·+ amCn−m

> D1(a1enλ1 + · · ·+ ame(n−m+1)λ1)

> D1e(n+1)λ1 .

However, this contradicts with the assumption that limn→∞1n log Cn = λ since λ1 > λ.

Hence we have ∑k≥1akekλ ≤ 1.

On the other hand, assume that ∑k≥1akekλ < 1. Then there exists some 0 < λ2 < λ such

that ∑k≥1akekλ < ∑k≥1

akekλ2

< 1.

Choose a constant D2 > 1 such that C0 < D2eλ2 . We can prove by induction that

Cn < D2e(n+1)λ2 for all n ≥ 0. Indeed, by induction hypothesis, (**) and ∑k≥1ak

ekλ2< 1, we

have

Cn < D2(a1enλ2 + · · ·+ aneλ2 + an+1) < D2e(n+1)λ2 .

This again contradicts with the assumption that limn→∞1n log Cn = λ since λ2 < λ.

This concludes the proof of the lemma.

4.4 Counterexample of Kikuta-Takahashi

Using Theorem 4.3.1, we can now construct counterexamples of Conjecture 4.1.3.

Proposition 4.4.1. For any even integer d ≥ 4, let X be a Calabi–Yau hypersurface in CPd+1 of

degree (d + 2) and Φ = TO (−⊗O(−1)). Then

h0(Φ) > 0 = log ρ(ΦH∗).

In particular, Conjecture 4.1.3 fails in this case.

120

Proof. By Theorem 4.3.1, we only need to show that the spectral radius ρ(ΦH∗) equals to 1.

Consider another autoequivalence Φ′ := TO (−⊗O(1)) on Db(X). By [7, Proposition

5.8], there is a commutative diagram

Db(X)Φ′ //

Ψ

Db(X)

Ψ

HMFgr(W)τ // HMFgr(W).

Here W is the defining polynomial of X, HMFgr(W) is the associated graded matrix factoriza-

tion category, Ψ is an equivalence introduced by Orlov [78], and τ is the grade shift functor

on HMFgr(W) which satisfies τd+2 = [2]. Hence we have (Φ′)d+2 = [2] and (Φ′)d+2H∗ = idH∗ .

On the other hand, (TO)H∗ is an involution on H∗(X; C) when X is an even dimensional

strict Calabi–Yau manifold ([51, Corollary 8.13]). Thus (TO)H∗ = (TO)−1H∗ . Hence we also

have Φd+2H∗ = idH∗ , which implies that ρ(ΦH∗) = 1.

Remark 4.4.2. The autoequivalence Φ′ that we considered in the proof is the one that corresponds to

the monodromy around the Gepner point (Zd+2-orbifold point) in the Kähler moduli of X.

Remark 4.4.3 (Ouchi). The functor Φ = TO (−⊗O(−1)) does not produce counterexamples

of Conjecture 4.1.3 if X is an odd dimensional Calabi–Yau manifold. When X is of odd dimension,

Lemma 4.3.2 implies

h0(Φ) = limn→∞

1n

log χ(G, Φn(G′)) ≤ log ρ([Φ]).

On the other hand, we have h0(Φ) ≥ log ρ([Φ]) by Kikuta-Shiraishi-Takahashi [62, Theorem 2.13].

4.5 Categorical entropy of P-twists

Despite the fact that we have disproved Conjecture 4.1.3, it still is an interesting problem

to find a characterization of autoequivalences satisfying the conjecture. The conjecture

holds true for autoequivalences on smooth projective curves [61], abelian surfaces [107], and

smooth projective varieties with ample (anti)-canonical bundle [63]. Ouchi [80] also showed

121

that spherical twists satisfy the conjecture.

By slightly modifying Ouchi’s proof, we show that Conjecture 4.1.3 also holds for

P-twists. One can consider P-twists as the categorical analogue of Dehn twists along

Lagrangian complex projective space.

We start with an easy lemma on the complexity function.

Lemma 4.5.1. 1. δt(G, E⊕ F) ≥ δt(G, E).

2. For a distinguished triangle A→ B→ C → A[1], we have

δt(G, B) ≤ δt(G, A) + δt(G, C).

We recall the notion of Pd-objects and Pd-twists introduced by Huybrechts and Thomas

[52]. They are the categorical analogue of the symplectomorphisms associated to a La-

grangian complex projective space [85].

Definition 4.5.2 ([52]). An object E ∈ Db(X) is called a Pd-object if E ⊗ ωX ∼= E and

Hom∗(E , E) is isomorphic as a graded ring to H∗(Pd, C).

For a Pd-object E , a generator h ∈ Hom2(E , E) can be viewed as a morphism h : E [−2]→

E . The image of h under the natural isomorphism Hom2(E , E) ∼= Hom2(E∨, E∨) will be

denoted h∨.

Definition 4.5.3 ([52]). The Pd-twist PE with respect to a Pd-object is an autoequivalence on

Db(X) given by the double cone construction

F 7→ PE (F ) := Cone(

Cone(Hom∗−2(E ,F )⊗ E → Hom∗(E ,F )⊗ E)→ F)

,

where the first right arrow is given by h∨ · id− id · h.

The following fact is crucial for computing the categorical entropy of P-twists in the

next section.

Lemma 4.5.4 ([52]). Let E ∈ Db(X) be a Pd-object. Then

1. PE (E) ∼= E [−2d].

122

2. PE (F ) ∼= F for any F ∈ E⊥ = F : Hom∗(E ,F ) = 0.

Theorem 4.5.5. Let X be a smooth projective variety of dimension 2d over C, and let PE be the

Pd-twist of a Pd-object E ∈ Db(X). Then for t ≤ 0, we have ht(PE ) = −2dt. In particular,

Conjecture 4.1.3 holds for Pd-twists:

h0(PE ) = 0 = log ρ((PE )H∗).

Assume that E⊥ 6= φ, then we have ht(PE ) = 0 for t > 0.

Proof. Fix a split generator G ∈ Db(X), and let

A := Cone(Hom∗−2(E , G)⊗ E → Hom∗(E , G)⊗ E).

By the definition of Pd-twist, we have a distinguished triangle

G → PE (G)→ A[1]→ G[1].

And by Lemma 4.5.4, we have PE (A) = A[−2d].

By applying Pn−1E to the distinguished triangle, we have

δt(G, PnE (G)) ≤ δt(G, Pn−1

E (G)) + δt(G, A[−2d(n− 1) + 1])

= δt(G, Pn−1E (G)) + δt(G, A)e(−2d(n−1)+1)t

≤ · · · (do this inductively)

≤ 1 + δt(G, A)n−1

∑k=0

e(−2dk+1)t

For t ≤ 0, we have δt(G, PnE (G)) ≤ 1 + δt(G, A) · ne(−2d(n−1)+1)t. Hence

ht(PE ) = limn→∞

1n

log δt(G, PnE (G)) ≤ −2dt.

On the other hand, since G⊕ E is also a split generator of D, we can apply Lemma 4.5.1

123

and 4.5.4 to get

ht(PE ) = limn→∞

1n

log δt(G, PnE (G⊕ E))

≥ limn→∞

1n

log δt(G, PnE (E))

= limn→∞

1n

log δt(G, E [−2nd])

= limn→∞

1n

log δt(G, E)e−2ndt

= −2dt.

Hence we have ht(PE ) = −2dt for t ≤ 0.

Note that the induced Fourier–Mukai type action (PE )H∗ is identity on the cohomology

[52]. Thus Conjecture 4.1.3 holds for the Pd-twist PE :

h0(PE ) = 0 = log ρ((PE )H∗).

For t > 0, we have δt(G, PnE (G)) ≤ 1 + δt(G, A)(et + n− 1). Hence ht(PE ) ≤ 0. Assume

that E⊥ 6= φ and take B ∈ E⊥, then we can apply the same trick on the split generator

G⊕ B:

ht(PE ) = limn→∞

1n

log δt(G, PnE (G⊕ B))

≥ limn→∞

1n

log δt(G, PnE (B))

= limn→∞

1n

log δt(G, B)

= 0.

This concludes the proof of the theorem.

Remark 4.5.6. When dim(X) = 2, a P1-object E ∈ Db(X) is also a spherical object (S2 ∼= P1).

For t ≤ 0, we have ht(PE ) = −2t. On the other hand, ht(TE ) = −t for the spherical twist TE [80].

This matches with the fact that T2E∼= PE by Huybrechts and Thomas [52].

124

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