Moduli spaces of Hecke modifications for rational and

msp Algebraic & Geometric Topology 21 (2021) 543–600

Moduli spaces of Hecke modificationsfor rational and elliptic curves

DAVID BOOZER

We propose definitions of complex manifolds PM .X;m; n/ that could potentiallybe used to construct the symplectic Khovanov homology of n–stranded links inlens spaces. The manifolds PM .X;m; n/ are defined as moduli spaces of Heckemodifications of rank 2 parabolic bundles over an elliptic curve X. To characterizethese spaces, we describe all possible Hecke modifications of all possible rank 2vector bundles over X, and we use these results to define a canonical open embeddingof PM .X;m; n/ into M s.X;mC n/ , the moduli space of stable rank 2 parabolicbundles over X with trivial determinant bundle and mC n marked points. We ex-plicitly compute PM .X; 1; n/ for nD 0; 1; 2 . For comparison, we present analogousresults for the case of rational curves, for which a corresponding complex manifoldPM .CP1; 3; n/ is isomorphic for n even to a space Y.S2; n/ defined by Seidel andSmith that can be used to compute the symplectic Khovanov homology of n–strandedlinks in S3 .

14H52, 14H99

1. Introduction 543

2. Terminology 548

3. Hecke modifications and parabolic bundles 548

4. Rational curves 557

5. Elliptic curves 570

6. Possible applications to topology 587

Appendix A. Vector bundles 589

Appendix B. Parabolic bundles 591

References 598

1 Introduction

Khovanov homology is a powerful invariant for distinguishing links in S3 [15]; it canbe viewed as a categorification of the Jones polynomial [12]: one can recover the Jones

Published: 25 April 2021 DOI: 10.2140/agt.2021.21.543

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544 David Boozer

polynomial of a link from its Khovanov homology, but the Khovanov homology gener-ally contains more information. For example, Khovanov homology can sometimes dis-tinguish distinct links that have the same Jones polynomial, and Khovanov homology de-tects the unknot — see Kronheimer and Mrowka [16] — but it is not known whether theJones polynomial has this property. The Khovanov homology of a link can be obtainedin a purely algebraic fashion by computing the homology of a chain complex constructedfrom a generic planar projection of the link. The Khovanov homology can also beobtained in a geometric fashion by Heegaard-splitting S3 into two 3–balls in such a waythat the intersection of the link with each 3–ball consists of r unknotted arcs. Each 3–ball determines a Lagrangian in a symplectic manifold Y.S2; 2r/, known as the Seidel–Smith space, and the Lagrangian Floer homology of the pair of Lagrangians yieldsthe Khovanov homology of the link (modulo grading); see Abouzaid and Smith [1]and Seidel and Smith [24]. Recently, Witten [29; 30; 31] has outlined gauge theoryinterpretations of Khovanov homology and the Jones polynomial in which the Seidel–Smith space is viewed as a moduli space of solutions to the Bogomolny equations.

Little is known about how Khovanov homology could be generalized to describe linksin 3–manifolds other than S3 , but such results would be of great interest. As a first steptowards this goal, one might consider the problem of generalizing Khovanov homologyto links in 3–manifolds with Heegaard genus 1, that is, lens spaces. In analogy with theSeidel–Smith approach to Khovanov homology, one could Heegaard-split a lens spaceinto two solid tori, each containing r unknotted arcs, and compute the Lagrangian Floerhomology of a pair of Lagrangians intersecting in a symplectic manifold Y.T 2; 2r/that generalizes the Seidel–Smith space Y.S2; 2r/.

In this paper we propose candidates for the manifold Y.T 2; 2r/. In outline, ourapproach is as follows. First, using a result due to Kamnitzer [13], we reinterpretthe Seidel–Smith space Y.S2; 2r/ as a moduli space H.CP1; 2r/ of equivalenceclasses of sequences of Hecke modifications of rank 2 holomorphic vector bundlesover a rational curve. Roughly speaking, a Hecke modification is a way of locallymodifying a holomorphic vector bundle near a point to obtain a new vector bundle.We show that there is a close relationship between Hecke modifications and parabolicbundles, and we use this relationship to reinterpret the Kamnitzer space H.CP1; 2r/

as a moduli space PM .CP1; 2r/ of isomorphism classes of parabolic bundles withmarking data. The space PM .CP1; 2r/ has two natural generalizations, PM .X; 1; 2r/and PM .X; 3; 2r/, to the case of an elliptic curve X, and we propose these spaces ascandidates for Y.T 2; 2r/.

Algebraic & Geometric Topology, Volume 21 (2021)

Moduli spaces of Hecke modifications for rational and elliptic curves 545

To explain our approach in detail, we first introduce some additional spaces. Given arank 2 holomorphic vector bundle E over a curve C, we define a set Htot.C;E; n/ ofequivalence classes of sequences of n Hecke modifications of E. As is well known,the set Htot.C;E; n/ has the structure of a complex manifold that is (noncanonically)isomorphic to .CP1/n , where each factor of CP1 corresponds to a single Hecke modi-fication. The Kamnitzer space H.CP1; 2r/ is then defined to be the open submanifoldof Htot.CP1;O˚O; 2r/ consisting of equivalence classes of sequences of Heckemodifications for which the terminal vector is semistable.

Example 1.1 For r D 1, we have that

Htot.CP1;O˚O; 2/D .CP1/2; H.CP1; 2/D .CP1/2�f.a; a/ j a 2CP1g:

To generalize the Kamnitzer space to curves of higher genus, we want to define modulispaces of sequences of Hecke modifications in which the initial vector bundle in thesequence is allowed to vary. We define such spaces via the use of parabolic bundles. Forany curve C and any natural numbers m and n, we define a moduli space P tot

M .C;m; n/

of marked parabolic bundles. We prove:

Theorem 1.2 The moduli space P totM .C;m; n/ naturally has the structure of a complex

manifold isomorphic to a .CP1/n–bundle over M s.C;m/.

Here the complex manifold M s.C;m/ is the moduli space of stable rank 2 parabolicbundles over a curve C with trivial determinant bundle and m marked points. Roughlyspeaking, the space P tot

M .C;m; n/ describes isomorphism classes of sequences of nHecke modifications in which the initial vector bundle in the sequence is allowed torange over isomorphism classes that are parametrized by M s.C;m/. By imposing acondition analogous to the semistability condition used to define the Kamnitzer spaceH.CP1; 2r/, we identify an open submanifold PM .C;m; n/ of P tot

M .C;m; n/. Weprove that P tot

M .CP1; n/ WDP totM .CP1; 3; n/ is isomorphic to Htot.CP1;O˚O; n/ and

PM .CP1; 2r/ WD PM .CP1; 3; 2r/ is isomorphic to H.CP1; 2r/. Thus the Seidel–Smith space Y.S2; 2r/, the Kamnitzer space H.CP1; 2r/ and the space of markedparabolic bundles PM .CP1; 2r/ are all isomorphic. However, unlike Y.S2; 2r/ orH.CP1; 2r/, the space PM .CP1; 2r/ naturally generalizes to the case of ellipticcurves.

Although our primary motivation for introducing P totM .CP1; n/ and PM .CP1; n/ is to

facilitate generalization, these spaces are also useful for proving canonical versions of


546 David Boozer

certain results for CP1 . For example, we prove a canonical version of the noncanonicalisomorphism Htot.CP1;O˚O; n/! .CP1/n for P tot

M .CP1; n/:

Theorem 1.3 There is a canonical isomorphism P totM .CP1; n/! .M ss.CP1; 4//n .

Here the complex manifold M ss.CP1; 4/Š CP1 is the moduli space of semistablerank 2 parabolic bundles over CP1 with trivial determinant bundle and 4 markedpoints. We also prove:

Theorem 1.4 There is a canonical open embedding

PM .CP1; m; n/!M s.CP1; mCn/:

Corollary 1.5 There is a canonical open embedding

PM .CP1; 2r/!M s.CP1; 2r C 3/:

Here the complex manifold M s.CP1; mCn/ is the moduli space of stable rank 2 para-bolic bundles over CP1 with trivial determinant bundle and mCn marked points. Forr D 1; 2 we have verified that the embedding of PM .CP1; 2r/ into M s.CP1; 2rC3/

agrees with a (noncanonical) embedding due to Woodward of the Seidel–Smith spaceY.S2; 2r/ into M s.CP1; 2r C 3/, and we conjecture that the agreement holds forall r .

We next proceed to the case of elliptic curves. We show that our reinterpretationPM .CP1; 2r/ of the Seidel–Smith space Y.S2; 2r/ has two natural generalizationsto the case of an elliptic curve X, namely PM .X; 1; 2r/ and PM .X; 3; 2r/. We provean elliptic-curve analog to Theorem 1.3:

Theorem 1.6 There is a canonical isomorphism P totM .X; 1; n/! .M ss.X//nC1 .

Here the complex manifold M ss.X/ŠCP1 is the moduli space of semistable rank 2vector bundles over an elliptic curve X with trivial determinant bundle. ComparingTheorems 1.3 and 1.6, we see that P tot

M .CP1; n/ is (noncanonically) isomorphic to.CP1/n , whereas P tot

M .X; 1; n/ is (noncanonically) isomorphic to .CP1/nC1 . Theextra factor of CP1 for P tot

M .X; 1; n/ can be understood from Theorem 1.2, whichstates that P tot

M .CP1; n/ D P totM .CP1; 3; n/ is a .CP1/n–bundle over M s.CP1; 3/

and P totM .X; 1; n/ is a .CP1/n–bundle over M s.X; 1/. But M s.CP1; 3/ is a single

point, whereas M s.X; 1/ is isomorphic to CP1 . We use Theorem 1.6 to explicitlycompute PM .X; 1; n/ for nD 0; 1; 2:



Theorem 1.7 The space PM .X; 1; n/ for nD 0; 1; 2 is given by

PM .X; 1; 0/DCP1;

PM .X; 1; 1/D .CP1/2�g.X/;

PM .X; 1; 2/D .CP1/3�f .X/;

where g WX! .CP1/2 and f WX! .CP1/3 are holomorphic embeddings defined inSections 5.5.2 and 5.5.3.

We also generalize the embedding result of Theorem 1.4 to the case of elliptic curves:

Theorem 1.8 There is a canonical open embedding PM .X;m; n/!M s.X;mCn/.

Here the complex manifold M s.X;mCn/ is the moduli space of stable rank 2 parabolicbundles on X with trivial determinant bundle and mCn marked points. In order toprove Theorems 1.6, 1.7 and 1.8, we construct a list of all possible Hecke modificationsof all possible rank 2 vector bundles on an elliptic curve X.

We conclude by discussing possible applications of our results to the problem ofgeneralizing symplectic Khovanov homology to lens spaces. We observe that theembedding results of Theorems 1.4 and 1.8 could be related to a conjectural spectralsequence from symplectic Khovanov homology to symplectic instanton homology,which would generalize a spectral sequence due to Kronheimer and Mrowka fromKhovanov homology to singular instanton homology [17]. Based on such considerations,we make the following conjectures:

Conjecture 1.9 The space PM .C; 3; 2r/ is the correct generalization of the Seidel–Smith space Y.S2; 2r/ to a curve C of arbitrary genus.

Conjecture 1.10 Given a curve C of arbitrary genus, there is a canonical openembedding PM .C;m; n/!M s.C;mCn/.

The paper is organized as follows. In Section 2 we introduce some terminology specificto this paper. In Section 3 we review the relevant background material on Heckemodifications, discuss the relationship between Hecke modifications and parabolicbundles, and define moduli spaces of marked parabolic bundles P tot

M .C;m; n/ andPM .C;m; n/. In Section 4 we describe Hecke modifications of rank 2 vector bundles onrational curves and show that the Seidel–Smith space Y.S2; 2r/ can be reinterpreted asthe space of marked parabolic bundles PM .CP1; 2r/DPM .CP1; 3; 2r/. In Section 5


548 David Boozer

we describe Hecke modifications of rank 2 vector bundles on elliptic curves anddiscuss possible elliptic-curve generalizations of the Seidel–Smith space. In Section 6we consider possible applications of our results to topology. In Appendices A and Bwe review the material we will need regarding vector bundles and parabolic bundles oncurves.

2 Terminology

Here we introduce some terminology specific to this paper.

Definition 2.1 Given a rank 2 holomorphic vector bundle E over a curve C, wedefine the instability degree of E to be degL � degE=L, where L � E is a linesubbundle of maximal degree.

The degree of the proper subbundles of a vector bundle E on a curve C is boundedabove (see for example [22, Lemma 3.21]), so the notion of instability degree is welldefined. The instability degree is positive for unstable bundles, 0 for strictly semistablebundles, and negative for stable bundles.

Definition 2.2 Given a rank 2 holomorphic vector bundle E over a curve C and apoint p 2 C, we say that a line p̀ 2 P .Ep/ in the fiber Ep over p is bad if there is aline subbundle L�E of maximal degree such that Lp D p̀ , and good otherwise.

Example 2.3 For the trivial bundle O˚O over CP1 , all lines are bad.

Definition 2.4 Given a rank 2 holomorphic vector bundle E over a curve C andpoints p1; : : : ; pn 2 C, we say that lines p̀i

2 P .Epi/ for i D 1; : : : ; n are bad in

the same direction if there is a line subbundle L � E of maximal degree such thatLpiD p̀i

for i D 1; : : : ; n.

3 Hecke modifications and parabolic bundles

3.1 Hecke modifications at a single point

A fundamental concept for us is the notion of a Hecke modification of a rank 2

holomorphic vector bundle. This notion is described in [13; 14]. Here we consider thecase of a single Hecke modification.

Definition 3.1 Let �E WE!C be a rank 2 holomorphic vector bundle over a curve C.A Hecke modification E ˛

p � F of E at a point p 2 C is a rank 2 holomorphic vector



bundle �F W F ! C and a bundle map ˛ W F ! E that satisfies the following twoconditions:

(1) The induced maps on fibers ˛q W Fq!Eq are isomorphisms for all points q 2Csuch that q ¤ p .

(2) We also impose a condition on the behavior of ˛ near p . We require thatthere is an open neighborhood U � C of p , local coordinates � W U ! V forV �C such that �.p/D 0, and local trivializations E W ��1E .U /!U �C2 and F W �

�1F .U /! U �C2 of E and F over U such that the following diagram

commutes:��1E .U / ��1F .U /

U �C2 U �C2

EŠ FŠ

˛

where the bottom horizontal arrow is

. E ı˛ ı �1F /.q; v/D

�q; x̨.�.q//v

�and x̨ W V !M.2;C/ has the form

x̨.z/D

�1 0

0 z

�:

It follows directly from Definition 3.1 that detF D .detE/˝O.�p/ and degF DdegE � 1. There is a natural notion of equivalence of Hecke modifications:

Definition 3.2 We say that two Hecke modifications E ˛

p � F and E ˛0

p � F 0 of

E at a point p 2 C are equivalent if there is an isomorphism � W F ! F 0 such that˛ D ˛0 ı� .

Definition 3.3 We define the total space of Hecke modifications Htot.C;EIp/ tobe the set of equivalence classes of Hecke modifications of a rank 2 vector bundle�E WE! C at a point p 2 C.

As is well known, the set Htot.C;EIp/ naturally has the structure of a complexmanifold that is (noncanonically) isomorphic to CP1 . A canonical version of thisstatement is:

Theorem 3.4 There is a canonical isomorphism

Htot.C;EIp/! P .Ep/; ŒE˛

p � F � 7! im. p̨ W Fp!Ep/:


550 David Boozer

It is also useful to think about Hecke modifications in terms of sheaves of sections.Consider a rank 2 vector bundle E and a line p̀ 2 P .Ep/. Let E be the sheaf ofsections of E, and define a subsheaf F of E whose set of sections over an open setU � C is given by

F.U /D fs 2 E.U / j p 2 U D) s.p/ 2 p̀g:

Define F to be the vector bundle whose sheaf of sections is F, and define ˛ W F !E

to be the bundle map corresponding to the inclusion of sheaves F ! E. ThenŒE

˛

p �F �2Htot.C;EIp/ corresponds to p̀ 2P .Ep/ under the isomorphism described

in Theorem 3.4. We have an exact sequence of sheaves

0! F ! E!Cp! 0;

where Cp is a skyscraper sheaf supported at the point p . It is important to note,however, that the usual notion of equivalence of extensions differs from the notion ofequivalence of Hecke modifications given in Definition 3.2.

3.2 Sequences of Hecke modifications

We would now like to generalize the notion of a Hecke modification of a vector bundleat a single point p 2 C to the notion of a sequence of Hecke modifications at distinctpoints .p1; : : : ; pn/ 2 C n .

Definition 3.5 Let �E WE ! C be a rank 2 holomorphic vector bundle over acurve C. A sequence of Hecke modifications E ˛1

p1 � E1

˛2

p2 � � � �

˛n

pn � En of E at

distinct points .p1; p2; : : : ; pn/ 2 C n is a collection of rank 2 holomorphic vectorbundles �Ei

WEi ! C and Hecke modifications Ei�1˛i

pi � Ei for i D 1; 2; : : : ; n,

where E0 WDE.

Definition 3.6 Two sequences of Hecke modifications E ˛1

p1 �E1

˛2

p2 � � � �

˛n

pn �En and

E˛01p1 �E 01

˛02p2 � � � �

˛0npn �E 0n are equivalent if there are isomorphisms �i WEi !E 0i such

that the following diagram commutes:

E E1 E2 � � � En

E E 01 E 02 � � � E 0n

D

˛1

�1Š

˛2

�2Š

˛3 ˛n

�nŠ

˛01 ˛02 ˛03 ˛0n



Definition 3.7 We define the total space of Hecke modifications Htot.C;EIp1; : : : ;pn/

to be the set of equivalence classes of sequences of Hecke modifications of the rank 2vector bundle �E WE! C at points .p1; : : : ; pn/ 2 C n . For simplicity, we will oftensuppress the dependence on p1; : : : ; pn and denote this space as Htot.C;E; n/.

Definition 3.8 We say that an isomorphism of vector bundles � WE ! E 0 is anisomorphism of equivalence classes of sequences of Hecke modifications

� W ŒE˛1

p1 �E1

˛2

p2 � � � �

˛n

pn �En�! ŒE 0

˛01p1 �E 01

˛02p2 � � � �

˛0npn �E 0n�

ifŒE 0

ˇ1

p1 �E1

˛2

p2 � � � �

˛n

pn �En�D ŒE

0 ˛01p1 �E 01

˛02p2 � � � �

˛0npn �E 0n�;

where ˇ1 WD � ı˛1 , or, equivalently, if there are isomorphisms �i WEi !E 0i such thatthe following diagram commutes:

E E1 E2 � � � En

E 0 E 01 E 02 � � � E 0n

�Š

˛1

�1Š

˛2

�2Š

˛3 ˛n

�nŠ

˛01 ˛02 ˛03 ˛0n

In what follows, it will be useful to reinterpret equivalence classes of sequences ofHecke modifications in terms of parabolic bundles. The relevant background materialon parabolic bundles is discussed in Appendix B. For our purposes here, a rank 2parabolic bundle over a curve C consists of a rank 2 holomorphic vector bundle�E WE! C, a parameter � > 0, called the weight, and a choice of line p̀i

2 P .Epi/

in the fiber EpiD ��1E .pi / over the point pi 2C for a finite number of distinct points

.p1; : : : ; pn/ 2 Cn . The data of just the marked points and lines, without the weight,

is referred to as a quasiparabolic structure on E. The additional data of the weightallows us to define the notions of stable, semistable and unstable parabolic bundles.

Definition 3.9 We define P tot.C;EIp1; : : : ; pn/D P .Ep1/� � � � �P .Epn

/ to be theset of all quasiparabolic structures with marked points .p1; : : : ; pn/ 2 C n on a rank 2holomorphic vector bundle �E WE ! C. For simplicity, we will often suppress thedependence on p1; : : : ; pn and denote this space as P tot.C;E; n/.

Since P .Epi/ is (noncanonically) isomorphic to CP1 , the set P tot.C;EIp1; : : : ; pn/

naturally has the structure of a complex manifold that is (noncanonically) isomorphicto .CP1/n . We have the following generalization of Theorem 3.4, which allows us toreinterpret Hecke modifications in terms of parabolic bundles:


552 David Boozer

Theorem 3.10 There is a canonical isomorphism Htot.C;E; n/!P tot.C;E; n/ givenby

ŒE˛1

p1 �E1

˛2

p2 � � � �

˛n

pn �En� 7! .E; p̀1

; : : : ; p̀n/;

where p̀iWD im..˛1 ı � � � ı˛i /pi

W .Ei /pi!Epi

/.

Under our reinterpretation of Hecke modifications in terms of parabolic bundles, anisomorphism of equivalence classes of sequences of Hecke modifications correspondsto an isomorphism of parabolic bundles:

Definition 3.11 We say that an isomorphism of vector bundles � WE!E 0 is an isomor-phism of parabolic bundles � W .E; p̀1

; : : : ; p̀n/! .E 0; `0p1

; : : : ; `0pn/ if �. p̀i

/D `0pi

for i D 1; : : : ; n.

Theorem 3.12 Two equivalence classes of sequences of Hecke modifications areisomorphic if and only if their corresponding parabolic bundles are isomorphic.

Proof This is a direct consequence of Theorem 3.10 and Definitions 3.8 and 3.11.

For many applications, given an equivalence class ŒE ˛1

p1 � E1

˛2

p2 � � � �

˛n

pn � En� of

sequences of Hecke modifications we will be interested only in the isomorphism classof the terminal vector bundle En , and it is useful to have a means of extracting thisinformation from the corresponding parabolic bundle .E; p̀1

; : : : ; p̀n/:

Definition 3.13 Let .E; p̀1; : : : ; p̀n

/ be a parabolic bundle over a curve C. Wedefine the Hecke transform H.E; p̀1

; : : : ; p̀n/ of E to be the vector bundle F that is

constructed as follows. Let E be the sheaf of sections of E. Define a subsheaf F of Ewhose set of sections over an open set U � C is given by

F.U /D fs 2 E.U / j pi 2 U D) s.pi / 2 p̀ifor i D 1; : : : ; ng:

Now define F to be the vector bundle whose sheaf of sections is F.

In particular, H.E; p̀1; : : : ; p̀n

/ is isomorphic to En . We will often want to pick outan open subset of P tot.C;E; n/ by using the Hecke transform to impose a semistabilitycondition:



Definition 3.14 Given a rank 2 holomorphic vector bundle E over a curve C and dis-tinct points .p1; : : : ; pn/2C n , we define P.C;E; n/ to be the subset of P tot.C;E; n/

consisting of parabolic bundles .E; p̀1; : : : ; p̀n

/ such that H.E; p̀1; : : : ; p̀n

/ issemistable. For simplicity, we are suppressing the dependence of P.C;m; n/ onp1; : : : ; pn in the notation.

Theorem 3.15 The set P.C;E; n/ is an open submanifold of P tot.C;E; n/.

Proof This follows from the fact that semistability is an open condition.

We have generalized the notion of a Hecke modification to the case of multiple pointsp1; : : : ; pn by considering sequences of Hecke modifications, for which the pointsmust be ordered. For most of our purposes we could equally well use an alternativegeneralization, described in [13], for which the points need not be ordered. Though wewill not use it here, we briefly describe this alternative generalization and show how itrelates to parabolic bundles:

Definition 3.16 Let �E WE ! C be a rank 2 holomorphic vector bundle over acurve C. A simultaneous Hecke modification E ˛

fp1;:::;png �� F of E at a set of distinct

points fp1; p2; : : : ; png � C is a rank 2 holomorphic vector bundle �F W F ! C anda bundle map ˛ W F !E that satisfies the following two conditions:

(1) The induced map on fibers ˛q WEq ! Fq is an isomorphism for all pointsq … fp1; : : : ; png.

(2) Condition (2) of Definition 3.1, which constrains the local behavior of ˛ near aHecke-modification point, holds at each of the points p1; : : : ; pn .

Definition 3.17 We say that two simultaneous Hecke modifications E ˛

fp1;:::;png �� F

and E ˛0

fp1;:::;png �� F 0 are equivalent if there is an isomorphism � W F ! F 0 such that

˛ D ˛0 ı� .

Definition 3.18 We define the total space of simultaneous Hecke modifications

Htot.C;E; fp1; : : : ; png/

to be the set of equivalence classes of simultaneous Hecke modifications of the rank 2vector bundle �E WE!C at the set of points fp1; : : : ; png�C. For simplicity, we willoften suppress the dependence on fp1; : : : ; png and denote this space as Htot.C;E; n/.


554 David Boozer

We can define a set of parabolic bundles P tot.C;E; n/ for which the marked points areunordered. We can define an isomorphism Htot.C;E; n/! P tot.C;E; n/ by

ŒE˛

fp1;:::;png �� F � 7! .E; p̀1

; : : : ; p̀n/;

where p̀iD im. p̨i

W Fpi!Epi

/. Since the Hecke transform does not depend on theordering of the points, it is well defined on parabolic bundles in P tot.C;E; n/, and wehave that H.E; p̀1

; : : : ; p̀n/ is isomorphic to F .

3.3 Moduli spaces of marked parabolic bundles

So far we have considered spaces of isomorphism classes of sequences of Heckemodifications in which the initial vector bundle in the sequence is held fixed. But inwhat follows we will want to generalize these spaces so the initial vector bundle isallowed to range over the isomorphism classes in a moduli space of vector bundles.Translating into the language of parabolic bundles, such spaces are equivalent to spacesof isomorphism classes of parabolic bundles in which the underlying vector bundlesare allowed to range over the isomorphism classes in a moduli space of vector bundles.However, there is a problem with defining such spaces arising from the fact that vectorbundles may have nontrivial automorphisms.

To illustrate the problem, consider the space P tot.CP1;O˚OIp1/, which is (non-canonically) isomorphic to CP1 . We might want to reinterpret this space as amoduli space of isomorphism classes of parabolic bundles of the form .E; p̀1

/ forŒE� 2M ss.CP1/, where M ss.CP1/, the moduli space of semistable rank 2 vectorbundles over CP1 with trivial determinant bundle, consists of the single point ŒO˚O�.But Aut.O˚O/DGL.2;C/, and for any pair of parabolic bundles .O˚O; p̀1

/ and.O˚O; `0p1

/ there is an automorphism � 2 Aut.O˚O/ such that �. p̀1/D `0p1

. Itfollows that all parabolic bundles of the form .O˚O; p̀1

/ are isomorphic and ourproposed moduli space collapses to a point, when what we wanted was CP1 .

To remedy the problem, we will add marking data to eliminate the nontrivial automor-phisms. In particular, since stable parabolic bundles have no nontrivial automorphisms,we make the following definition:

Definition 3.19 Given a curve C and distinct points .q1; : : : ; qm;p1; : : : ;pn/2CmCn,we define the total space of marked parabolic bundles P tot

M .C;m; n/ to be the set ofisomorphism classes of parabolic bundles of the form .E; `q1

; : : : ; `qm; p̀1

; : : : ; p̀n/



such that ŒE; `q1; : : : ; `qm

� 2M s.C;m/:

P totM .C;m; n/D fŒE; `q1

; : : : ; `qm; p̀1

; : : : ; p̀n� j ŒE; `q1

; : : : ; `qm� 2M s.C;m/g:

For simplicity, we are suppressing the dependence of P totM .C;m; n/ on q1; : : : ; qm and

p1; : : : ; pn in the notation.

Here the complex manifold M s.C;m/ is the moduli space of stable rank 2 parabolicbundles over C with trivial determinant bundle and m marked points. We will referto the lines `q1

; : : : ; `qmas marking lines, since their purpose is to add additional

structure to E so as to eliminate nontrivial automorphisms. We will refer to the lines

p̀1; : : : ; p̀n

as Hecke lines, since their purpose is to parametrize Hecke modificationsat the points p1; : : : ; pn . Because we have defined P tot

M .C;m; n/ in terms of stableparabolic bundles, which have no nontrivial automorphisms, the collapsing phenomenondescribed above does not occur, and we have the following result:

Theorem 3.20 The set P totM .C;m; n/ naturally has the structure of a complex manifold

isomorphic to a .CP1/n–bundle over M s.C;m/.

The base manifold M s.C;m/ constitutes the moduli space over which the isomorphismclasses of vector bundles with marking data range, and the .CP1/n fibers correspond toa space of Hecke modifications CP1 for each of the points p1; : : : ; pn . We will proveTheorem 3.20 by constructing P tot

M .C;m; n/ from a universal CP1–bundle, which wefirst describe for the case mD 0:

Lemma 3.21 There is a universal CP1–bundle P ! C �M s.C /, which has theproperty that for any complex manifold S and any CP1–bundle Q ! C � S, thebundle Q is isomorphic to the pullback of P along 1C � fQ for a unique mapfQ W S !M s.C /.

Lemma 3.21 is proven in [5]. One way to understand this result is as follows. LetM sr;d.C / denote the moduli space of stable vector bundles of rank r and degree d on a

curve C. As discussed in [9], one can define a corresponding moduli stack Bunsr;d.C /

and a Gm–gerbe � W Bunsr;d.C /! Hom.�;M s

r;d.C //; this is a morphism of stacks

for which all the fibers are isomorphic to BGm . The stack Hom.�; C /�Bunsr;d.C /

carries a universal rank r vector bundle E. One can show (see [8, Corollary 3.12]) that ifgcd.r; d/D 1 then M s

r;d.C / is a fine moduli space and E descends to a universal vector

bundle E! C �M sr;d.C /, which can be viewed as a generalization of the Poincaré


556 David Boozer

line bundle L! C � Jac.C / for the case r D 1 and d D 0. By projectivizing E, wealso get a universal CP r�1–bundle P .E/! C �M s

r;d.C /. If gcd.r; d/ ¤ 1, then

M sr;d.C / is not a fine moduli space and C �M s

r;d.C / does not carry a universal vector

bundle. It is still possible, however, to use E to construct a universal CP r�1–bundleP ! C �M s

r;d.C /, only now this CP r�1–bundle is not the projectivization of a

universal vector bundle. One way to make this result plausible is to note that whereas astable vector bundle has automorphism group C� , consisting of automorphisms thatscale the fibers by a constant factor, the projectivization of a stable vector bundle hastrivial automorphism group, consisting of just the identity automorphism.

Similar results hold for moduli spaces of stable vector bundles for which the determinantbundle is a fixed line bundle. In particular, the space M s.C / of stable rank 2 vectorbundles with trivial determinant bundle is not a fine moduli space and C �M s.C / doesnot carry a universal vector bundle; nonetheless, it does carry a universal CP1–bundleP ! C �M s.C /. We will use this universal CP1–bundle to construct P tot

M .C;m; n/

for the case mD 0:

Proof of Theorem 3.20 First consider the case m D 0, and note that M s.C; 0/ D

M s.C /. Given a point p 2 C, let P.p/!M s.C / denote the pullback of the uni-versal CP1–bundle P ! C �M s.C / described in Lemma 3.21 along the inclusionM s.C /! C �M s.C /, ŒE� 7! .p; ŒE�/. Given distinct points .p1; : : : ; pn/ 2 C n ,we can pull back the .CP1/n–bundle P.p1/� � � � �P.pn/! .M s.C //n along thediagonal map M s.C /! .M s.C //n , ŒE� 7! .ŒE�; : : : ; ŒE�/ to obtain P tot

M .C; 0; n/.

The proof for m > 0 is the same. One can define a moduli stack corresponding toM s.C;m/ that carries a universal rank 2 parabolic bundle [9]. Using a numericalcondition analogous to the condition gcd.r; d/D 1 for vector bundles [9, Example 5.7;6, Proposition 3.2], one can show that M s.C;m/ is a fine moduli space for m> 0 andthe universal parabolic bundle on the moduli stack descends to a universal parabolicbundle on C �M s.C;m/. We can projectivize this latter bundle and use it to constructP totM .C;m; n/ in the same manner as for the mD 0 case.

We will often want to pick out an open subset of P totM .C;m; n/ by imposing a semista-

bility condition:

Definition 3.22 Given a curve C and distinct points .q1; : : : ; qm; p1; : : : ; pn/ 2CmCn , we define PM .C;m; n/ to be the subset of P tot

M .C;m; n/ consisting of points



ŒE; `q1; : : : ; `qm

; p̀1; : : : ; p̀n

� such that H.E; p̀1; : : : ; p̀n

/ is semistable. For sim-plicity, we are suppressing the dependence of PM .C;m; n/ on q1; : : : ; qm; p1; : : : ; pnin the notation.

Theorem 3.23 The set PM .C;m; n/ is an open submanifold of P totM .C;m; n/.

Proof This follows from the fact that semistability is an open condition.

We can interpret marked parabolic bundles in terms of Hecke modifications as follows:

Definition 3.24 We define a sequence of Hecke modifications of a parabolic bundle.E; `q1

; : : : ; `qm/ to be a sequence of Hecke modifications of the underlying vector

bundle E.

Definition 3.25 We say that two sequences of Hecke modifications

.E; `q1; : : : ; `qm

/˛1

p1 �E1

˛2

p2 � � � �

˛n

pn �En;

.E 0; `0q1; : : : ; `0qm

/˛01p1 �E 01

˛02p2 � � � �

˛0npn �E 0n

are equivalent if there are isomorphisms �i WEi !E 0i for i D 0; : : : ; n such that thefollowing diagram commutes:

.E; `q1; : : : ; `qm

/ E1 E2 � � � En

.E 0; `0q1; : : : ; `0qm

/ E 01 E 02 � � � E 0n

�0Š

˛1

�1Š

˛2

�2Š

˛3 ˛n

�nŠ

˛01 ˛02 ˛03 ˛0n

The space P totM .C;m; n/ can then be interpreted as a moduli space of equivalence classes

of sequences of Hecke modifications of parabolic bundles, and the space PM .C;m; n/can be interpreted as the subspace P tot

M .C;m; n/ consisting of equivalence classes ofsequences for which the terminal vector bundles are semistable. We will not use theseinterpretations here, since it is simpler to work directly with the marked parabolicbundles.

4 Rational curves

4.1 Vector bundles on rational curves

Grothendieck showed that all rank 2 holomorphic vector bundles on (smooth projective)rational curves are decomposable [7]; that is, they have the form O.n/˚O.m/ for


558 David Boozer

integers n and m. The instability degree of O.n/˚O.m/ is jn�mj, so the bundleO.n/˚O.m/ is strictly semistable if n D m and unstable otherwise. There are nostable rank 2 vector bundles on rational curves.

4.2 List of all possible single Hecke modifications

Here we present a list of all possible Hecke modifications at a point p 2CP1 of allpossible rank 2 vector bundles on CP1 . We will parametrize Hecke modifications of avector bundle E at a point p in terms of lines p̀ 2P .Ep/, as described in Theorem 3.4.Since we are always free to tensor a Hecke modification with a line bundle, it sufficesto consider vector bundles of nonnegative degree.

Theorem 4.1 Consider the vector bundle O.n/˚O for n � 1 (unstable, instabilitydegree n). The possible Hecke modifications are

O.n/˚O �O.n/˚O.�1/ if p̀ DO.n/p (a bad line),O.n� 1/˚O otherwise (a good line).

Proof (1) ( p̀ D O.n/p ) A Hecke modification ˛ WO.n/˚O.�1/! O.n/˚Ocorresponding to p̀ is

˛ D

�1 0

0 f

�;

where f WO.�1/!O is the unique (up to rescaling by a constant) nonzero morphismsuch that fp D 0 on the fibers over p .

(2) ( p̀¤O.n/p ) Since n�1, we can choose a section t of O.n/ such that t .p/¤0.Choose a section sD .at; b/ of O.n/˚O for a; b 2C such that s.p/¤ 0 and s 2 p̀ .A Hecke modification ˛ WO.n� 1/˚O!O.n/˚O corresponding to p̀ is

˛ D

�f at

0 b

�;

where f WO.n � 1/! O.n/ is the unique (up to rescaling by a constant) nonzeromorphism such that fp D 0 on the fibers over p .

Theorem 4.2 Consider the vector bundle O ˚ O (strictly semistable, instabilitydegree 0). The possible Hecke modifications are

O˚O O˚O.�1/ for all p̀ (all lines are bad ):



Proof Define a section s D .a; b/ of O˚O for a; b 2 C such that s.p/ ¤ 0 ands.p/2 p̀ . If bD 0, then a Hecke modification ˛ WO˚O.�1/!O˚O correspondingto p̀ is

˛ D

�a 0

0 f

�;

and if b¤ 0, then a Hecke modification ˛ WO˚O.�1/!O˚O corresponding to p̀

is˛ D

�a f

b 0

�;

where f WO.�1/!O is the unique (up to rescaling by a constant) nonzero morphismsuch that fp D 0 on the fibers over p .

4.2.1 Observations From this list, we make the following observations:

Lemma 4.3 The following results hold for Hecke modifications of a rank 2 vectorbundle E over CP1 :

(1) A Hecke modification of E changes the instability degree by ˙1.

(2) Hecke modification of E corresponding to a line p̀ 2 P .Ep/ changes theinstability degree by �1 if p̀ is a good line and C1 if p̀ is a bad line.

(3) A generic Hecke modification of E changes the instability degree by �1 unlessE has the minimum possible instability degree 0, in which case all Heckemodifications of E change the instability degree by C1.

4.3 Moduli spaces P totM

.CP 1; m; n/ and PM .CP 1; m; n/

Our goal is to define a moduli space of Hecke modifications that is isomorphic to theSeidel–Smith space Y.S2; 2r/. Kamnitzer showed that such a space can be defined asfollows:

Definition 4.4 (Kamnitzer [13]) Given distinct points .p1; : : : ; p2r/ 2 .CP1/2r ,define the Kamnitzer space H.CP1; 2r/ to be the subset of Htot.CP1;O˚O; 2r/consisting of equivalence classes of sequences of Hecke modifications

O˚O ˛1

p1 �E1

˛2

p2 � � � �

˛2r

p2r ��E2r

such that E2r is semistable.

In particular, the condition that E2r is semistable implies that E2r DO.�r/˚O.�r/.


560 David Boozer

Theorem 4.5 (Kamnitzer [13]) The Kamnitzer space H.CP1; 2r/ has the structureof a complex manifold isomorphic to the Seidel–Smith space Y.S2; 2r/.

We will describe the Seidel–Smith space Y.S2; 2r/ and Kamnitzer’s isomorphism inSection 4.6. We can use the results of Section 3.2 to reinterpret the Kamnitzer spaceH.CP1; 2r/ in terms of parabolic bundles:

Definition 4.6 Define P tot.CP1; n/ WD P tot.CP1;O ˚ O; n/ and P.CP1; 2r/ WD

P.CP1;O˚O; 2r/.

Theorem 4.7 There is a canonical isomorphism H.CP1; 2r/! P.CP1; 2r/.

Proof This follows from restricting the domain and range of the canonical isomorphismHtot.CP1;O˚O; 2r/! P tot.CP1;O˚O; 2r/ described in Theorem 3.10.

We can also reinterpret the spaces P tot.CP1; n/ and P.CP1; 2r/ in terms of themoduli spaces of marked parabolic bundles that we defined in Section 3.3. In whatfollows, we will choose a global trivialization O˚O! CP1 �C2 and identify allthe fibers of O˚O with C2 . We can then identify lines p̀ 2 P .Ep/ with pointsin CP1 and speak of lines in different fibers as being equal or unequal. In Appendix Awe define a moduli space M ss.CP1/ of semistable rank 2 vector bundles with trivialdeterminant bundle, and in Appendix B we define a moduli space M s.CP1; m/ ofstable rank 2 parabolic bundles with trivial determinant bundle and m marked points.From the fact that M ss.CP1/D fŒO˚O�g and Aut.O˚O/D GL.2;C/, we obtainthe following results:

Theorem 4.8 The moduli space M s.CP1; 3/ consists of the single point

ŒO˚O; `q1; `q2

; `q3�;

where `q1, `q2

and `q3are any three distinct lines. Given any two stable parabolic

bundles of the form .O˚O; `q1; `q2

; `q3/ and .O˚O; `0q1

; `0q2; `0q3

/, there is a unique(up to rescaling by a constant) automorphism � 2 Aut.O˚O/ such that �.`qi

/D `0qi

for i D 1; 2; 3.

Corollary 4.9 There is an isomorphism M s.CP1; 3/!M ss.CP1/ given by

ŒO˚O; `q1; `q2

; `q3� 7! ŒO˚O�:

These results motivate the following definitions of “marked” versions of P tot.CP1; n/

and P.CP1; 2r/:



Definition 4.10 We define P totM .CP1; n/ WD P tot

M .CP1; 3; n/ and PM .CP1; n/ WD

PM .CP1; 3; n/.

The marked and unmarked versions of these spaces are easily seen to be isomorphic:

Theorem 4.11 The spaces P totM .CP1; n/ and P tot.CP1; n/ are (noncanonically) iso-

morphic.

Proof Choose distinct lines `q1, `q2

and `q3, and define an isomorphism

P tot.CP1; n/! P totM .CP1; n/

byŒO˚O; p̀1

; : : : ; p̀n� 7! ŒO˚O; `q1

; `q2; `q3

; p̀1; : : : ; p̀n

�:

The isomorphism is not canonical, since it depends on the choice of lines `q1, `q2

and `q3.

Theorem 4.12 The spaces PM .CP1; n/ and P.CP1; n/ are (noncanonically) iso-morphic.

Proof This follows from restricting the domain and range of the isomorphism

P tot.CP1; n/! P totM .CP1; n/

described in Theorem 4.11.

Our primary motivation for defining P totM .CP1; n/ is to draw a parallel with the case

of elliptic curves, which we consider in Section 5. But the space P totM .CP1; n/ also

has an advantage over P tot.CP1; n/ in that we can use the marking lines to rendercertain constructions canonical. For example, we can define a canonical version of thenoncanonical isomorphism P tot.CP1; n/D P tot.CP1;O˚O; n/! .CP1/n :

Lemma 4.13 Fix a parabolic bundle .E; `q1; `q2

; `q3/ such that ŒE; `q1

; `q2; `q3

� 2

M s.CP1; 3/ and a point p 2 CP1 . There is a canonical isomorphism P .Ep/ !

M ss.CP1; 4/ŠCP1 given by

p̀ 7! ŒE; `q1; `q2

; `q3; p̀�:

Proof This follows from the fact that .E; `q1; `q2

; `q3/ is stable, so the lines `q1

, `q2

and `q3are all distinct under the global trivialization of E.


562 David Boozer

Theorem 4.14 There is a canonical isomorphism h W P totM .CP1; n/!.M ss.CP1; 4//n .

Proof Define maps hi W P totM .CP1; n/!M ss.CP1; 4/ for i D 1; : : : ; n by

hi .ŒE; `q1; `q2

; `q3; p̀1

; : : : ; p̀n�/D ŒE; `q1

; `q2; `q3

; p̀i�:

Then h WD .h1; : : : ; hn/ is an isomorphism by Theorem 3.20 and Lemma 4.13.

Remark 4.15 Definition 3.22 for PM .C;m; n/ implies that PM .CP1; m; n/D¿ forodd n, since there are no semistable rank 2 vector bundles of odd degree on CP1 . Wecould alternatively define PM .CP1; m; n/ by requiring that H.E; p̀1

; : : : ; p̀n/ have

the minimal possible instability degree, which is 0 for n even and 1 for n odd. Thiscondition is equivalent to semistability for n even, but is a distinct condition for n odd,and gives a nonempty space.

4.4 Embedding PM .CP 1; m; n/ ! M s.CP 1; m C n/

We will now describe a canonical open embedding of the space PM .CP1; m; n/ intothe space of stable parabolic bundles M s.CP1; mCn/. We first need two lemmas:

Lemma 4.16 Given a parabolic bundle .O˚O; p̀1; : : : ; p̀n

/ over CP1 , if the lines

p̀1; : : : ; p̀n

are bad in the same direction then H.O˚O; p̀1; : : : ; p̀n

/DO˚O.�n/.

Proof Since p̀1; : : : ; p̀n

are bad in the same direction, we have that p̀1D � � � D p̀n

under a global trivialization of O˚O in which all the fibers are identified with C2 .An explicit sequence of Hecke modifications with p̀1

D � � � D p̀nis given by

O˚O ˛1

p1 �O˚O.�1/ ˛2

p2 � � � �

˛n

pn �O˚O.�n/;

where O˚O ˛1

p1 �O˚O.�1/ is a Hecke modification corresponding to the line p̀1

and for i D 2; : : : ; n we define

˛i D

�1 0

0 fi

�;

where fi WO.�i/!O.�i C 1/ is the unique (up to rescaling by a constant) nonzeromorphism such that .fi /pi

D 0 on the fibers over pi .

Lemma 4.17 Given a parabolic bundle

.O˚O; p̀1; : : : ; p̀n

/

over CP1 , if H.O ˚ O; p̀1; : : : ; p̀n

/ is semistable then .O ˚ O; p̀1; : : : ; p̀n

/ issemistable.



Proof We will prove the contrapositive, so assume that .O ˚ O; p̀1; : : : ; p̀n

/ isunstable. It follows that more than 1

2n of the lines are bad in the same direction. Let

s denote the number of such lines, and choose a permutation � 2 †n such that thefirst s points of .�.p1/; : : : ; �.pn// correspond to these lines. By Lemma 4.16 wehave that H.O˚O; `�.p1/; : : : ; `�.ps//DO˚O.�s/, which has instability degree s .Lemma 4.3 states that a single Hecke modification changes the instability degree by ˙1,so H.O˚O; `�.p1/; : : : ; `�.pn//DH.O˚O; p̀1

; : : : ; p̀n/ has instability degree at

least s� .n� s/D 2s�n > 0, and is thus unstable.

Remark 4.18 The converse to Lemma 4.17 does not always hold; for example, con-sider the semistable parabolic bundle .O ˚ O; p̀1

; p̀2; p̀3

; p̀4/ for points pi D

Œ1 W�i � 2CP1 , where

p̀1D Œ1 W 0�; p̀2

D Œ0 W 1�;

p̀3D Œ1 W 1�; p̀4

D Œ.�3��1/.�4��2/ W .�3��2/.�4��1/�:

One can show that H.O˚O; p̀1; p̀2

; p̀3; p̀4

/DO.�3/˚O.�1/, which is unstable.

Theorem 4.19 There is a canonical open embedding

PM .CP1; m; n/!M s.CP1; mCn/:

Proof Take ŒE; `q1; : : : ; `qm

; p̀1; : : : ; p̀n

� 2 PM .CP1; m; n/; note that E DO˚O.Since .E; `q1

; : : : ; `qm/ is stable, fewer than 1

2m of the lines `q1

; : : : ; `qmare equal

under the global trivialization of E. Since H.E; p̀1; : : : ; p̀n

/ is semistable, it followsfrom Lemma 4.17 that .E; p̀1

; : : : ; p̀n/ is semistable, so at most 1

2n of the lines

p̀1; : : : ; p̀n

are equal under the global trivialization of E. It follows that fewer than12.mCn/ of the lines `q1

; : : : ; `qm; p̀1

; : : : ; p̀nare equal, so

.E; `q1; : : : ; `qm

; p̀1; : : : ; p̀n

/

is stable. So PM .CP1; m; n/ is a subset of M s.CP1; mC n/. Specifically, the setPM .CP1; m; n/ consists of points ŒE; `q1

; : : : ; `qm; p̀1

; : : : ; p̀n�2M s.CP1; mCn/

such that .E; `q1; : : : ; `qm

/ is stable and H.E; p̀1; : : : ; p̀n

/ is semistable. Sincestability and semistability are open conditions, we have that PM .CP1; m; n/ is anopen subset of M s.CP1; mCn/.

4.5 Examples

We can generalize the Kamnitzer space H.CP1; n/ to allow for both even and odd n,in analogy with the generalization described in Remark 4.15:


564 David Boozer

Definition 4.20 Given distinct points .p1; : : : ; pn/ 2 .CP1/n , define the Kamnitzerspace H.CP1; n/ to be the subset of Htot.CP1;O˚O; n/ consisting of equivalenceclasses of sequences of Hecke modifications O˚O ˛1

p1 �E1

˛2

p2 � � � �

˛n

pn �En such that

En has the minimum possible instability degree (0 for n even, 1 for n odd.)

Here we compute Kamnitzer space H.CP1; n/ for nD 0; 1; 2; 3.

4.5.1 Calculate H.CP1; 0/ We have

H.CP1; 0/DHtot.CP1; 0/D fO˚Og:

4.5.2 Calculate H.CP1; 1/ All Hecke modifications of O˚O give O˚O.�1/,which has instability degree 1, so

H.CP1; 1/DHtot.CP1; 1/DCP1:

4.5.3 Calculate H.CP1; 2/ A sequence of two Hecke modifications of O˚O musthave one of two forms:

O˚O ˛1

p1 �O˚O.�1/ ˛2

p2 �O.�1/˚O.�1/;

O˚O ˛1

p1 �O˚O.�1/ ˛2

p2 �O˚O.�2/:

In the first case the terminal bundle O.�1/˚O.�1/ is semistable, whereas in the secondcase the terminal bundle O˚O.�2/ is unstable. So H.CP1; 2/ is the complement inHtot.CP1; 2/ of sequences of Hecke modifications of the second form. As we showedin the proof of Lemma 4.16, the resulting space is

H.CP1; 2/D .CP1/2�f.a; a/ j a 2CP1g:

4.5.4 Calculate H.CP1; 3/ Now consider a sequence of three Hecke modificationsof O˚O. The only sequences for which the terminal bundle does not have instabilitydegree 1 are of the form

O˚O ˛1

p1 �O˚O.�1/ ˛2

p2 �O˚O.�2/ ˛3

p3 �O˚O.�3/:

So H.CP1; 3/ is the complement in Htot.CP1; 3/ of sequences of Hecke modificationsof this form. As we showed in the proof of Lemma 4.16, the resulting space is

H.CP1; 3/D .CP1/3�f.a; a; a/ j a 2CP1g:



4.6 The Seidel–Smith space Y.S 2; 2r/

Here we compare the embedding of PM .CP1; 2r/ into M s.CP1; 2r C 3/ that wedefined in Theorem 4.19 with an embedding of the Seidel–Smith space Y.S2; 2r/ intoM s.CP1; 2r C 3/ due to Woodward. We begin by defining the Seidel–Smith spaceY.S2; 2r/.

Definition 4.21 We define the Slodowy slice S2r to be the subspace of gl.2r;C/

consisting of matrices with 2� 2 identity matrices I on the superdiagonal, arbitrary2� 2 matrices in the left column and zeros everywhere else.

Example 4.22 Elements of S6 have the form0@ Y1 I 0

Y2 0 I

Y3 0 0

1A ;where Y1 , Y2 and Y3 are arbitrary 2� 2 complex matrices.

Definition 4.23 Define a map � W S2r !C2r=†2r that sends a matrix to the multisetof the roots of its characteristic polynomial, where a root of multiplicity m occurs mtimes in the multiset.

Definition 4.24 Given distinct points .�1; : : : ; �2r/ 2C2r , define the Seidel–Smithspace Y.S2; 2r/ to be the fiber ��1.f�1; : : : ; �2rg/. For simplicity, we are suppressingthe dependence of Y.S2; 2r/ on �1; : : : ; �2r in the notation. This space was introducedin [24], which denotes Y.S2; 2r/ by Yr .

The Seidel–Smith space Y.S2; 2r/ naturally has the structure of a complex manifold,in fact a smooth complex affine variety. In what follows, it will be useful to definelocal coordinates � W U ! V on CP1 , where U D fŒ1 W z� j z 2 Cg � CP1 , V D C

and �.Œ1 W z�/D z . We define points pi WD ��1.�i / 2 CP1 corresponding to �i fori D 1; : : : ; 2r .

4.6.1 Kamnitzer isomorphism H.CP1; 2r/ ! Y.S 2; 2r/ Here we describe anisomorphism due to Kamnitzer from the space of Hecke modifications H.CP1; 2r/ tothe Seidel–Smith space Y.S2; 2r/.


566 David Boozer

Define global meromorphic sections sn of O.n/ such that div sn D n � Œ1�. For eachrank 2 vector bundle E DO.n/˚O.m/, define standard meromorphic sections

e1E D .sn; 0/; e2E D .0; sm/;

and define a standard local trivialization E W ��1E .U /! U �C2 of E over U by

e1E .p/ 7! .p; .1; 0//; e2E .p/ 7! .p; .0; 1//:

Consider an element of H.CP1; 2r/,

(1) ŒE0˛1

p1 �E1

˛2

p2 � � � �

˛2r

p2r ��E2r �;

where E0 DO˚O. Define rank 2 free CŒz�–modules Li for i D 0; : : : ; 2r as spacesof sections of Ei over U,

Li D �.U;Ei /DCŒz� � fe1Ei; e2Eig:

The sequence of Hecke modifications (1) then yields a sequence of CŒz�–modulemorphisms x̨i ,

L0x̨1 � L1

x̨2 � � � �

x̨2r � L2r :

We can also view x̨i as a holomorphic map x̨i W V !M.2;C/, defined as in Definition3.1, such that

. Ei�1ı˛i ı

�1Ei/.q; v/D

�q; x̨i .�.q//v

�:

Define an 2r –dimensional complex vector space V by

V D coker.x̨1 ı x̨2 ı � � � ı x̨2r/D L0=.x̨1 ı x̨2 ı � � � ı x̨2r/.L2r/:

One can show that an ordered basis for V is given by

.zr�1e1E0; zr�1e2E0

; : : : ; ze1E0; ze2E0

; e1E0; e2E0

/:

Note that z acts C–linearly on V , and thus defines a 2r�2r complex matrix A relativeto this basis.

Theorem 4.25 (Kamnitzer [13]) We have an isomorphism H.CP1; 2r/!Y.S2; 2r/given by

ŒE0˛1

p1 �E1

˛2

p2 � � � �

˛2r

p2r ��E2r � 7! A:

To perform calculations, it is useful to have explicit expressions for the maps x̨i . Foreach vector bundle E and point p D Œ1 W�� 2 U, we use the standard trivialization



E W ��1.E/! U �C2 to identify P .Ep/ with CP1 . For each line p̀ 2 P .Ep/D

CP1 , we give a holomorphic map x̨ W V !M.2;C/ that describes a Hecke modification˛ W F !E corresponding to p̀ :

Hecke modifications of O.n/˚O for n� 1:

p̀ D Œ1 W 0� W O.n/˚O ˛

p �O.n/˚O.�1/; x̨.z/D

�1 0

0 z��

�;

p̀ D Œ� W 1� W O.n/˚O ˛

p �O.n� 1/˚O; x̨.z/D

�z��

0 1

�:

Hecke modifications of O˚O :

p̀ D Œ1 W 0� W O˚O ˛

p �O˚O.�1/; x̨.z/D

�1 0

0 z��

�;

p̀ D Œ� W 1� W O˚O ˛

p �O˚O.�1/; x̨.z/D

�� z��

1 0

�:

4.6.2 Woodward embedding Y.S 2; 2r/!M s.CP1; 2rC3/ Here we describe anembedding due to Woodward (unpublished notes) of the Seidel–Smith space Y.S2; 2r/into the space of stable rank 2 parabolic bundles M s.CP1; 2rC3/. We first make thefollowing definition:

Definition 4.26 Given distinct points .p1; : : : ; pn/ 2 .CP1/n , define a subspacePss.CP1; n/ of P tot.CP1;O ˚ O; n/ consisting of semistable parabolic bundles.O˚O; p̀1

; : : : ; p̀n/.

In particular, Pss.CP1; n/ consists of parabolic bundles .O˚O; p̀1; : : : ; p̀n

/ forwhich at most 1

2n of the lines are equal to any given line under a global trivialization of

O˚O. Given distinct points .q1; q2; q3; p1; : : : ; pn/ 2 .CP1/nC3 and distinct lines`q1; `q2

; `q32CP1 , we can define an embedding Pss.CP1; n/!M s.CP1; nC 3/,

.O˚O; p̀1; : : : ; p̀n

/ 7! ŒO˚O; `q1; `q2

; `q3; p̀1

; : : : ; p̀n�:

We will define an embedding Y.S2; 2r/! Pss.CP1; 2r/. Composing with the em-bedding Pss.CP1; 2r/!M s.CP1; 2rC3/ will then yield the Woodward embedding.We first define some vectors. Define vectors x; y 2C2 by

x D .1; 0/; y D .0; 1/:


568 David Boozer

Define vectors x1; : : : ; xr ; y1; : : : ; yr 2C2r by

x1 D .x; 0; : : : ; 0/; x2 D .0; x; 0; : : : ; 0/; : : : ; xr D .0; : : : ; 0; x/;

y1 D .y; 0; : : : ; 0/; y2 D .0; y; 0; : : : ; 0/; : : : ; yr D .0; : : : ; 0; y/:

Define vectors x.�/; y.�/ 2C2r by

x.�/D .�r�1x; �r�2x; : : : ; �x; x/D �r�1x1C�r�2x2C � � �C�xr�1C xr ;

y.�/D .�r�1y; �r�2y; : : : ; �y; y/D �r�1y1C�r�2y2C � � �C�yr�1Cyr :

We use the vectors to define a subspace W.s; t/ of C2r , and we calculate its dimension:

Definition 4.27 Given .s; t/ 2C2 , define a subspace

W.s; t/DC � fsx.�/C ty.�/ j � 2Cg

of C2r .

Lemma 4.28 For .s; t/ 2C2�f0g, we have that dimW.s; t/D r .

Proof Define a vector w.s; t; �/ 2C2r by

(2) w.s; t; �/ WD sx.�/C ty.�/D

rXjD1

�r�j .sxj C tyj /:

Let S �C2r denote the span of the linearly independent vectors

fsx1C ty1; : : : ; sxr C tyrg:

Clearly W.s; t/ � S. Form an r � r matrix V whose i th row vector consists of thecomponents of w.s; t; i/ relative to the ordered basis .sx1C ty1; : : : ; sxr C tyr/ of S.From (2), it follows that the .i; j / matrix element of V is given by

Vij D .i/r�j :

So V is a Vandermonde matrix corresponding to the distinct integers .1; 2; : : : ; r/, andthus has nonzero determinant. It follows that the vectors fw.s; t; 1/; : : : ; w.s; t; r/g arelinearly independent, hence W.s; t/D S and dimW.s; t/D dimS D r .

We are now ready to define the Woodward embedding. Take a matrix A 2 Y.S2; 2r/.Let v.�/ 2C2r be a left eigenvector of A with eigenvalue �:

v.�/AD �v.�/:

Given the form of A, it follows that

v.�/DX.�/x.�/CY.�/y.�/



for some X.�/; Y.�/ 2C . Since A 2 Y.S2; 2r/, the eigenvalues of A are

�1; : : : ; �2r 2C:

Define lines p̀i2CP1 for i D 1; : : : ; 2r by

p̀iD ŒX.�i / WY.�i /�:

Theorem 4.29 (Woodward) We have an embedding Y.S2; 2r/! Pss.CP1; 2r/,A 7! .O˚O; p̀1

; : : : ; p̀2r/.

Proof A priori the codomain of the map is P tot.CP1; 2r/, so we need to showthat the image is in fact contained in Pss.CP1; 2r/. Note that if p̀i

D Œs W t � thenv.�i / 2 W.s; t/. Since the eigenvalues �1; : : : ; �2r are distinct, the eigenvectorsfv.�1/; : : : ; v.�2r/g are linearly independent, so the maximum number of eigen-vectors that can live in W.s; t/ is dimW.s; t/ D r by Lemma 4.28. So at most rof the lines p̀1

; : : : ; p̀2rcan be equal to any given line Œs W t � in CP1 , and thus

.O˚O; p̀1; : : : ; p̀2r

/ is semistable.

Lemma 4.17 states that we have an embedding P.CP1; 2r/! Pss.CP1; 2r/. Wecan precompose this embedding with the canonical isomorphism H.CP1; 2r/ !

P.CP1; 2r/ described in Theorem 4.7 to obtain an embedding

H.CP1; 2r/! Pss.CP1; 2r/;

and we obtain a commutative diagram

H.CP1; 2r/ Pss.CP1; 2r/

PM .CP1; 2r/ M ss.CP1; 2r C 3/

Š

where the bottom horizontal arrow is the embedding described in Theorem 4.19. It isinteresting to compare the embedding H.CP1; 2r/!Pss.CP1; 2r/ to the embeddingY.S2; 2r/! Pss.CP1; 2r/ from Theorem 4.29. We make the following conjecture:

Conjecture 4.30 There is a commutative diagram

H.CP1; 2r/ Pss.CP1; 2r/

Y.S2; 2r/ Pss.CP1; 2r/

Š Š


570 David Boozer

where the left downward arrow is the Kamnitzer isomorphism and the right downwardarrow is the map on parabolic bundles induced by � 2Aut.O˚O/DGL.2;C/, where

� D

�0 �1

1 0

�:

Theorem 4.31 Conjecture 4.30 holds for r D 1 and r D 2.

Proof This can be shown by a direct calculation.

5 Elliptic curves

5.1 Vector bundles on elliptic curves

Vector bundles on elliptic curves have been classified by Atiyah [2]:

Definition 5.1 Define E.r; d/ to be the set of isomorphism classes of indecomposablevector bundles of rank r and degree d on an elliptic curve X.

The set E.r; d/ naturally has the structure of a complex manifold, and we have thefollowing result:

Theorem 5.2 (Atiyah [2]) There are isomorphisms Jac.X/ ! E.r; d/ for all rand d .

In particular, E.1; d/ is the set of isomorphism classes of line bundles of degree d ,and the isomorphism Jac.X/! E.1; d/ is given by ŒL� 7! ŒL˝O.d � e/� for a choiceof basepoint e 2X. Here we summarize the facts we will need regarding line bundlesand rank 2 vector bundles on elliptic curves. Results that are well known will be statedwithout proof; full proofs can be found in [2; 11; 26].

Definition 5.3 We say that a degree 0 line bundle L is 2–torsion if L2 DO.

There are four 2–torsion line bundles on an elliptic curve. We will denote the 2–torsionline bundles by Li for i D 1; 2; 3; 4, with the convention that L1 DO.

Definition 5.4 Given line bundles L and M on an elliptic curve, an extension of Lby M is an exact sequence

0!M !E! L! 0;

where E is a rank 2 vector bundle.



Lemma 5.5 Given line bundles L and M on an elliptic curve , equivalence classes ofextensions of L by M are classified by Ext1.L;M/DH 0.L˝M�1/.

Lemma 5.6 (Teixidor [26, Lemma 4.5]) If ŒE� 2 E.2; d/, then h0.E/D 0 if d < 0and h0.E/D d if d > 0, where h0.E/ WD dimH 0.E/.

We will now list the rank 2 vector bundles on an elliptic curve X. Up to tensoring witha line bundle, we have the following vector bundles:

5.1.1 Rank 2 decomposable vector bundles Decomposable bundles have the formL1˚L2 , where L1 and L2 are line bundles. The instability degree of L1˚L2 isjdegL1� degL2j, so L1˚L2 is strictly semistable if degL1 D degL2 and unstableotherwise. The proof of the following result is straightforward:

Lemma 5.7 Let EDL1˚L2 , where L1 and L2 are line bundles such that degL1>degL2 . At a point p 2X the line .L1/p is bad , and all other lines in P .Ep/ are good.

A semistable decomposable bundle must have even degree, so after tensoring with asuitable line bundle it has the form L˚L�1 , where L is a degree 0 line bundle. Thereare two subclasses of such bundles: the four bundles Li ˚Li , and bundles L˚L�1

such that L2 ¤ O. These two subclasses of semistable decomposable bundles havevery different properties:

Lemma 5.8 The bundle Li ˚Li has no good lines , and Aut.Li ˚Li /D GL.2;C/.

Lemma 5.9 Let E D L˚L�1 , where L is a degree 0 line bundle such that L2 ¤O.The automorphism group Aut.E/ is the subgroup of GL.2;C/ matrices of the form�

A 0

0 D

�:

At a point p 2X the lines Lp and .L�1/p are bad , and all other lines in P .Ep/ aregood. Given a pair of good lines p̀; `

0p 2 P .Ep/, there is a unique (up to rescaling by

a constant) automorphism � 2 Aut.E/ such that �. p̀/D �.`0p/.

The proofs of Lemmas 5.8 and 5.9 are straightforward and have been omitted.

5.1.2 Rank 2 degree 0 indecomposable bundles There is a unique indecomposablebundle F2 that can be obtained via an extension of O by O,

(3) 0!O ˛�! F2

ˇ�!O! 0:


572 David Boozer

The bundle F2 is strictly semistable, and hence has instability degree 0. The mapJac.X/! E.2; 0/, ŒL� 7! ŒF2˝L�, is an isomorphism, so in particular F2˝LD F2if and only if LDO.

Lemma 5.10 If L is a degree 0 line bundle , then

Hom.L; F2/D�

C �˛ if LDO,0 otherwise ,

Hom.F2; L/D�

C �ˇ if LDO,0 otherwise.

Proof Apply Hom.L;�/ to the short exact sequence (3) to obtain the long exactsequence

0! Hom.L;O/ ˛��! Hom.L; F2/ˇ��! Hom.L;O/ ı

�! Ext1.L;O/:

If L¤O, then Hom.L;O/D 0 and the long exact sequence implies Hom.L; F2/D 0.So assume LDO. Then Hom.L;O/DC � 1O . To prove that Hom.L; F2/DC �˛ , itsuffices to show that ı.1O/¤ 0. Assume for contradiction that this is not the case. Thenthe long exact sequence implies that there is a morphism f 2 Hom.L; F2/ such thatˇ�.f /Dˇıf D 1O . It follows that the short exact sequence (3) splits, a contradiction.

The claim regarding Hom.F2; L/ can be proven in a similar manner by applyingHom.�; L/ to the short exact sequence (3).

Lemma 5.11 Given a point q 2X, there are nonzero sections t0 and t1 of F2˝O.q/such that

(1) H 0.F2˝O.q//DC � ft0; t1g;

(2) div t0 D 0 and div t1 D q ;

(3) t1.p/ 2O.q/p for all p 2X, where O.q/! F2˝O.q/ is the unique degree 1line subbundle of F2˝O.q/;

(4) ft0.p/; t1.p/g are linearly independent for all p 2X such that p ¤ q .

Proof Tensoring ˛ WO! F2 with O.q/ and precomposing with the unique (up torescaling by a constant) nonzero morphism O ! O.q/, we obtain a section t1 ofF2˝O.q/ such that div t1 D q and t1.p/ 2O.q/p for all p 2X. By Lemma 5.6 wehave that h0.F2˝O.q//D 2, so we can choose a section t0 of F2˝O.q/ linearlyindependent from t1 .

We claim that div t0 D 0. Assume for contradiction that this is not the case. We obtaina subbundle O.div t0/!F2˝O.q/, and by semistability of F2˝O.q/ it follows that



div t0 D p for some p 2X. We thus obtain a subbundle O.p/! F2˝O.q/, hence asubbundle O.p� q/! F2 . But this contradicts Lemma 5.10 unless p D q , in whichcase t0 and t1 are linearly dependent.

We claim t0.p/ and t1.p/ are linearly independent at all points p 2 X such thatp ¤ q . Assume for contradiction that they are linearly dependent at some point pdistinct from q . Then we can choose a nonzero section s D at0C bt1 of F2˝O.q/for a; b 2C such that s.p/D 0. We thus obtain a subbundle O.div s/! F2˝O.q/.We have that p 2 div s , so semistability of F2˝O.q/ implies that div s D p . We thusobtain a subbundle O.p/! F2˝O.q/, hence a subbundle O.p� q/! F2 . But thiscontradicts Lemma 5.10.

Lemma 5.12 The automorphism group Aut.F2/ is the subgroup of GL.2;C/ matri-ces of the form �

A B

0 A

�:

At a point p 2X the line Op is bad , where O! F2 is the unique degree 0 subbundleof F2 , and all other lines in P ..F2/p/ are good. Given a pair of good lines p̀; `

0p 2

P ..F2/p/, there is a unique (up to rescaling by a constant) automorphism � 2 Aut.E/such that �. p̀/D �.`0p/.

Proof Apply Hom.�; F2/ to the short exact sequence (3) to obtain

(4) 0! Hom.O; F2/ˇ��! Hom.F2; F2/

˛��! Hom.O; F2/:

Note that ˛�.1F2/ D ˛ , so Lemma 5.10 implies that ˛� is surjective and thus the

sequence (4) is in fact short exact. It follows that Hom.F2; F2/DC � f1F2; �g, where

� WD ˇ�.˛/ D ˛ ı ˇ . Note that � ı � D 0, so we can define an injective grouphomomorphism Aut.F2/! GL.2;C/ by

A1F2CB� 7!

�A B

0 A

�:

The fact that Op is the unique bad line of .F2/p follows from Lemma 5.10. Given goodlines p̀; `

0p 2 P ..F2/p/, choose nonzero vectors v; v0; w 2 .F2/p such that v 2 p̀ ,

v0 2 `0p and w 2Op . Since p̀¤Op , it follows that fv;wg is a basis for .F2/p . Definea; b 2C such that v0D avCbw ; note that since `0p ¤Op we have that a¤ 0. Definec 2 C such that �p.v/ D cw ; note that since �p.w/ D 0 and �p ¤ 0, we have thatc ¤ 0. Then v0 D �p.v/, where � D a1F C .b=c/� 2 Aut.F2/. Hence �. p̀/D `0p ,and � is clearly unique up to rescaling by a constant.


574 David Boozer

5.1.3 Rank 2 degree 1 indecomposable bundles Given a point p 2 X, there is aunique degree 1 indecomposable bundle G2.p/ that can be obtained via an extensionof O.p/ by O,

0!O!G2.p/!O.p/! 0:

The bundle G2.p/ is stable, with instability degree �1. The map E.1; 1/! E.2; 1/,ŒO.p/� 7! ŒG2.p/�, is an isomorphism, with inverse given by det W E.2; 1/! E.1; 1/,ŒE�! ŒdetE�. It follows that for any degree 0 divisor D on X we have that

G2.pC 2D/DG2.p/˝O.D/;

and, in particular, G2.p/˝LŠG2.p/ if and only if L2 DO.

Lemma 5.13 We have that Aut.G2.p//DC� consists only of trivial automorphismsthat scale the fibers by a constant factor.

Proof This follows from the fact that G2.p/ is stable.

Lemma 5.14 Any degree 0 line bundle L is a subbundle of G2.p/ via a unique (upto rescaling by a constant) inclusion map L!G2.p/.

Proof Let L be a degree 0 line bundle. By Lemma 5.6 we have that

h0.G2.p/˝L�1/D 1;

hence G2.p/˝L�1 has a nonzero section s . We thus obtain a subbundle O.div s/!G2.p/˝ L

�1 . By stability of G2.p/˝ L�1 , we must have div s D 0. Tensoringwith L, we obtain a subbundle L!G2.p/. The claim regarding uniqueness followsfrom the fact that h0.G2.p/˝L�1/D 1.

Corollary 5.15 All lines of G2.p/ are bad.

Proof This is shown in Theorem 5.19.

5.2 List of all possible single Hecke modifications

Here we present a list of all possible Hecke modifications at a point p 2 X of allpossible rank 2 vector bundles on X, up to tensoring with a line bundle. We willparametrize Hecke modifications of a vector bundle E at a point p in terms of lines

p̀ 2 P .Ep/, as described in Theorem 3.4. Since we are always free to tensor a Heckemodification with a line bundle, it suffices to consider vector bundles of nonnegativedegree.



To construct the list, we will often use the following strategy. By tensoring E witha line bundle of sufficiently high degree if necessary, we can assume without lossof generality that E is generated by global sections. Consider a Hecke modification˛ W F ! E of E at p corresponding to a line p̀ WD im p̨ 2 P .Ep/. Since we haveassumed E is generated by global sections, there is a section s of E such that s.p/¤ 0and s.p/ 2 p̀ . We then get a subbundle O.div s/!E and a commutative diagram

0 O.div s/ F L˝O.�p/ 0

0 O.div s/ E L 0

˛ f

where f is the unique (up to rescaling by a constant) such nonzero morphism. ThusF is an extension of L˝O.�p/ by O.div s/, and we can often use this informationto determine F .

5.2.1 Rank 2 bundles of degree greater than 1

Theorem 5.16 Consider a bundle of the form L˚O for L a line bundle of degreegreater than 1 (unstable , instability degree degL). The possible Hecke modificationsare

L˚O �L˚O.�p/ if p̀ D Lp (a bad line),.L˝O.�p//˚O otherwise (a good line).

Proof (1) ( p̀DLp ) A Hecke modification ˛ W L˚O.�p/!L˚O correspondingto p̀ is

˛ D

�1 0

0 f

�;

where f is the unique (up to rescaling by a constant) nonzero morphism O.�p/!O.

(2) ( p̀ ¤ Lp ) Since degL > 1, we can choose a nonzero section t of L suchthat t .p/ ¤ 0. Since t is nonvanishing at p , we can choose a section s D .at; b/of L˚O for a; b 2 C such that s.p/ ¤ 0 and s.p/ 2 p̀ . A Hecke modification˛ W .L˝O.�p//˚O! L˚O corresponding to p̀ is

˛ D

�f at

0 b

�;

where f is the unique (up to rescaling by a constant) nonzero morphism

L˝O.�p/! L:


576 David Boozer

5.2.2 Rank 2 bundles of degree 1

Theorem 5.17 Consider the bundle O.q/ ˚ O with q ¤ p (unstable, instabilitydegree 1). The possible Hecke modifications are

O.q/˚O �O.q/˚O.�p/ if p̀ DO.q/p (a bad line),O.q�p/˚O otherwise (a good line).

Proof One can prove this result by using the fact that O.q/ has a section t suchthat t .p/ ¤ 0 and writing down explicit Hecke modifications, as in the proof ofTheorem 5.16.

Theorem 5.18 Consider the bundle O.p/˚O (unstable, instability degree 1). Thepossible Hecke modifications are

O.p/˚O

8<:O.p/˚O.�p/ if p̀ DO.p/p (a bad line),O˚O if p̀ DOp (a good line),F2 otherwise (a good line).

Proof (1) ( p̀ D O.p/p ) A Hecke modification ˛ WO.p/˚O.�p/! O.p/˚Ois given by

˛ D

�1 0

0 f

�;

where f is the unique (up to rescaling by a constant) nonzero morphism O.�p/!O.

(2) ( p̀ DOp ) A Hecke modification ˛ WO˚O!O.p/˚O is given by

˛ D

�t 0

0 1

�;

where t is the unique (up to rescaling by a constant) nonzero morphism O!O.p/.

(3) ( p̀ ¤ O.p/p and p̀ ¤ Op ) Pick a point q 2 X such that q ¤ p . Choose anonzero section t0 of O.pCq/ such that t0.q/¤ 0 and t0.p/¤ 0. Choose a nonzerosection t1 of O.q/. Note that div t1 D q . Since t0.p/ ¤ 0 and t1.p/ ¤ 0, we candefine a section s D .at0; bt1/ of O.pC q/˚O.q/ for a; b 2C such that s.p/¤ 0and s.p/ 2 p̀ . Since p̀ ¤ O.p/p and p̀ ¤ Op , it follows that a ¤ 0 and b ¤ 0,thus div s D 0 and we obtain a subbundle O.div s/DO!O.pC q/˚O.q/, 1 7! s .Thus we have a commutative diagram

0 O F O.2q/ 0

0 O O.pC q/˚O.q/ O.2qCp/ 0

˛



The bundle F cannot split, since there are no nonzero morphisms

O.2q/!O.pC q/˚O.q/;

hence F is indecomposable. Since detF DO.2q/ it follows that F DF2˝O.q/˝Lfor a 2–torsion line bundle L. We can compose ˛ with projection onto the secondsummand of O.pCq/˚O.q/ to obtain a nonzero morphism F!O.q/, so Lemma 5.10implies LDO and F D F2˝O.q/.

Theorem 5.19 Consider the bundle G2.p/ (stable , instability degree �1). There is acanonical isomorphism P .G2.p/p/!M ss.X/ŠCP1 given by

p̀ 7! ŒH.G2.p/; p̀/�:

All lines of G2.p/ are bad.

Proof By Lemma 5.14, any degree 0 line bundle L is a subbundle of G2.p/ via aunique (up to rescaling by a constant) inclusion map L! G2.p/. Thus we have acommutative diagram

0 L F L�1 0

0 L G2.p/ L�1˝O.p/ 0

˛

Note that Ext1.L�1; L/ D H 0.L�2/. If L2 ¤ O then H 0.L�2/ D 0, so F splits,thus F D L˚L�1 .

Now suppose L2 DO. We claim that F is indecomposable; assume for contradictionthat this is not the case. Then F D L˚L, so ˛ W F !G2.p/ gives a map O˚O!G2.p/˝L that is an isomorphism away from p , so we obtain two linearly independentsections of G2.p/˝ L. But by Lemma 5.6 we have that h0.G2.p/˝ L/ D 1, acontradiction. It follows that F is indecomposable. Since detF DO, it follows thatF D F2 ˝M for a 2–torsion line bundle M. Since we have a nonzero morphismL! F , Lemma 5.10 implies that M D L and F D F2˝L.

The above considerations show that we have a surjection Jac.X/!M ss.X/, ŒL� 7! ŒF �.The vector bundle F is isomorphic to H.G2.p/; p̀/, where p̀ 2P .G2.p/p/ is the linecorresponding to ŒG2.p/

˛

p �F �2Htot.X;G2.p/Ip/ under the canonical isomorphism


578 David Boozer

described in Theorem 3.4, and we have a commutative diagram

Jac.X/ M ss.X/

P .G2.p/p/

Here Jac.X/ ! P .G2.p/p/ is given by ŒL� 7! Lp and P .G2.p/p/ ! M ss.X/ isgiven by p̀ 7! ŒH.G2.p/; p̀/�. Since Jac.X/!M ss.X/ is surjective, we have thatJac.X/! P .G2.p/p/ is surjective and P .G2.p/p/! M ss.X/ is an isomorphism.The surjectivity of Jac.X/! P .G2.p/p/ implies that all lines of P .G2.p/p/ are bad.Since G2.p/DG2.q/˝M for a suitable degree 0 line bundle M, all lines of G2.p/are bad.

5.2.3 Rank 2 bundles of degree 0

Theorem 5.20 Consider the bundle O˚O (strictly semistable , instability degree 0).The possible Hecke modifications are

O˚O O˚O.�p/ for all p̀ (all lines are bad ).

Proof We can choose a section s of O˚O such that s.p/¤ 0 and s D p̀ . We thusobtain a subbundle O!O˚O, 1 7! s and a commutative diagram

0 O F O.�p/ 0

0 O O˚O O 0

˛

Since Ext1.O.�p/;O/DH 0.O.�p//D0, we have that F splits, thus F DO˚O.�p/.Alternatively, one can write down explicit Hecke modifications, as in the proof ofTheorem 5.16.

Theorem 5.21 Consider a bundle of the form L˚L�1 , where L is a degree 0 linebundle such that L2¤O (strictly semistable , instability degree 0). The possible Heckemodifications are

L˚L�1

8̂<̂:L˚ .L�1˝O.�p// if p̀ D Lp (a bad line),.L˝O.�p//˚L�1 if p̀ D .L

�1/p (a bad line),G2.p/˝O.�p/ otherwise (a good line).



Proof For p̀DLp or p̀D .L�1/p , we can write down explicit Hecke modifications,

as in the proof of Theorem 5.16. So assume p̀ ¤ Lp and p̀ ¤ .L�1/p . Choose a

point e 2 X such that .L˚L�1/˝O.e/ D O.q1/˚O.q2/ for points q1; q2 2 Xdistinct from p . Since L2 ¤O, it follows that q1 ¤ q2 . Note that q1C q2 D 2e . Lettk be the unique (up to rescaling by a constant) nonzero section of O.qk/; note thatdiv tkDqk . We can define a section sD .at1; bt2/ of O.q1/˚O.q2/ for a; b 2C suchthat s.p/¤ 0 and s.p/ 2 p̀ . Since p̀ ¤ Lp and p̀ ¤ .L

�1/p , it follows that a¤ 0and b¤ 0, thus div sD 0. We thus obtain a subbundle O.div s/DO!O.q1/˚O.q2/,1 7! s , and a commutative diagram

0 O F O.q1C q2�p/ 0

0 O O.q1/˚O.q2/ O.q1C q2/ 0

˛

There are no nonzero morphisms O.q1Cq2�p/!O.q1/˚O.q2/, so F cannot split.Since detF DO.q1Cq2�p/, we have that F DG2.q1Cq2�p/DG2.2.e�p/Cp/DG2.p/˝O.e�p/.

Theorem 5.22 Consider the bundle F2 (strictly semistable , instability degree 0). Thepossible Hecke modifications are

F2

�O˚O.�p/ if p̀ DOp (a bad line),G2.p/˝O.�p/ otherwise (a good line),

where O! F2 is the unique degree 0 line subbundle of F2 .

Proof (1) ( p̀ DOp ) We have a commutative diagram

0 O F O.�p/ 0

0 O F2 O 0

˛

Since Ext1.O.�p/;O/DH 0.O.�p//D0, we have that F splits, thus F DO˚O.�p/.

(2) ( p̀ ¤ Op ) Pick a point q 2 X such that q ¤ p . Choose sections t0 and t1 ofF2˝O.q/ as in Lemma 5.11. We can define a section s D at0C bt1 of F2˝O.q/for a; b 2C such that s.p/¤ 0 and s.p/ 2 p̀ . Since p̀ ¤Op it follows that a¤ 0,thus div s D 0. We thus obtain a subbundle O.div s/DO! F2˝O.q/, 1 7! s , and


580 David Boozer

a commutative diagram

0 O F O.2q�p/ 0

0 O F2˝O.q/ O.2q/ 0

˛

We claim that F cannot split. Assume for contradiction that F splits, thus F DO ˚ O.2q � p/. Then we can precompose ˛ with the inclusion O.2q � p/ !O˚O.2q�p/ to obtain a nonzero morphism O.2q�p/!F2˝O.q/, contradictingLemma 5.10. Since F does not split and detF D O.2q � p/, we have that F DG2.2q�p/DG2.2.q�p/Cp/DG2.p/˝O.q�p/.

5.2.4 Observations From this list, we make the following observations:

Lemma 5.23 The following results hold for Hecke modifications of a rank 2 vectorbundle E on an elliptic curve:

(1) A Hecke modification of E changes the instability degree by ˙1.

(2) Hecke modification of E corresponding to a line p̀ 2 P .Ep/ changes theinstability degree by �1 if p̀ is a good line and C1 if p̀ is a bad line.

(3) A generic Hecke modification of E changes the instability degree by �1 unlessE has the minimum possible instability degree �1, in which case all Heckemodifications of E change the instability degree by C1.

5.3 Moduli spaces P totM

.X; m; n/ and PM .X; m; n/

In Section 4.3 we defined a total space of marked parabolic bundles P totM .CP1; n/D

P totM .CP1; 3; n/ for rational curves, and we showed that the Seidel–Smith space

Y.S2; 2r/ could be reinterpreted as the subspace PM .CP1; 2r/DPM .CP1; 3; 2r/ ofP totM .CP1; 2r/. We now want to generalize the spaces P tot

M .CP1; n/ and PM .CP1; 2r/

to the case of an elliptic curve X. There are obvious candidates: namely, the spacesP totM .X;m; n/ and PM .X;m; n/ for some value of m, which should be chosen to obtain

the correct generalization. One possibility is to use same value mD 3 that we used forrational curves. But mD 1 also yields a reasonable generalization, as can be understoodfrom the following considerations.

In Appendix A we define a moduli space M ss.C / of semistable rank 2 vector bundlesover a curve C with trivial determinant bundle, and in Appendix B we define a



moduli space M s.C;m/ of stable rank 2 parabolic bundles over a curve C with trivialdeterminant bundle and m marked points. Recall that for rational curves we chosem D 3 marking lines because we wanted P tot

M .CP1; m; n/ to be isomorphic to thespace of parabolic bundles P tot.CP1;O˚O; n/ in which the underlying vector bundleis O˚O, and

P totM .CP1; 3; 0/DM s.CP1; 3/DM ss.CP1/D fŒO˚O�g:

For an elliptic curve X, however, the corresponding spaces M s.X; 3/ and M ss.X/ arenot isomorphic: the space M s.X; 3/ is a complex manifold of dimension 3, whereasM ss.X/ is isomorphic to CP1 . Instead we have the following results, which can beviewed as elliptic-curve analogs to Theorem 4.8 and Corollary 4.9 for rational curves:

Theorem 5.24 The moduli space M s.X; 1/ consists of points ŒE; `q�, where `q isa good line and either E D F2˝Li or E D L˚L�1 for L a degree 0 line bundlesuch that L2 ¤O. Given any two parabolic bundles of the form .E; `q/ and .E; `0q/

representing points of M s.X; 1/, there is a unique (up to rescaling by a constant)automorphism � 2 Aut.E/ such that �.`q/D `0q .

Proof If ŒE; `q� 2M s.X; 1/ then E is semistable, detE DO and `q is a good line.Since E is semistable and detE DO, it must be Li ˚Li , F2˝Li , or L˚L�1 forL a degree 0 line bundle such that L2¤O. But Lemma 5.8 states that Li˚Li has nogood lines, so E cannot be Li ˚Li . Lemmas 5.9 and 5.12 show that the remainingtwo possibilities for E do have good lines and also prove the statement regardingunique automorphisms.

Corollary 5.25 The map M s.X; 1/!M ss.X/, ŒE; `q� 7! ŒE�, is an isomorphism.

From these results, we see that there are two natural generalizations of P totM .CP1; n/

to an elliptic curve. The generalization of P totM .CP1; 3; 0/DM ss.CP1/ is

P totM .X; 1; 0/DM

s.X; 1/DM ss.X/;

which would lead us to choose mD 1 marking lines. On the other hand, the general-ization of P tot

M .CP1; 3; 0/DM s.CP1; 3/ is

P totM .X; 3; 0/DM

s.X; 3/;

which would lead us to choose mD 3 marking lines. We will address the question ofwhich of these values of m yields the correct generalization of the Seidel–Smith spacein Section 6.


582 David Boozer

From Theorem 3.20, we have that P totM .X; 1; n/ is a .CP1/n–bundle over M s.X; 1/Š

CP1 . We will show that this bundle is trivial. To prove this result, we will use themarking line of P tot

M .X; 1; n/ to canonically identify P .Ep/ with M ss.X/ŠCP1 forŒE; `q1

; p̀1; : : : ; p̀n

� 2 P totM .X; 1; n/:

Lemma 5.26 Fix a parabolic bundle .E; `q/ such that ŒE; `q� 2M s.X; 1/, a pointp 2X such that p¤ q , and a point e 2X such that pCq D 2e . There is a canonicalisomorphism P .Ep/!M ss.X/ given by

p̀ 7! ŒH.E; `q; p̀/˝O.e/�:

Proof Theorem 5.24 implies that `q is a good line and either E D F2 ˝ Li orE D L˚L�1 for L a degree 0 line bundle such that L2 ¤O. From Theorems 5.21and 5.22, it follows that

H.E; `q/DG2.q/˝O.�q/DG2.pC 2.q� e//˝O.�q/DG2.p/˝O.�e/:

The result now follows from Theorem 5.19.

Lemma 5.26 can be viewed as the elliptic-curve analog to Lemma 4.13 for rationalcurves. To perform calculations, it will be useful to explicitly evaluate the map P .Ep/!

M ss.X/ for bad lines p̀ 2 P .Ep/. In general, we prove:

Lemma 5.27 Fix distinct points p; q 2 X and a point e 2 X such that pC q D 2e .If `q is a good line , then

H.L˚L�1; `q; Lp/˝O.e/DM ˚M�1; where M D L˝O.p� e/;

H.L˚L�1; `q; .L�1/p/˝O.e/DM ˚M�1; where M D L˝O.q� e/;

H.L˚L�1; Lq; .L�1/p/˝O.e/DM ˚M�1; where M D L˝O.q� e/;

H.F2; `q;Op/˝O.e/DM ˚M�1; where M DO.p� e/:

Proof These results are straightforward calculations using the list of Hecke modifi-cations in Section 5.2. As an example, we will prove the result involving F2 . FromTheorem 5.22 we have that

H.F2;Op/DO˚O.�p/:

Since `q is a good line, the bundle H.F2;Op; `q/DH.F2; `q;Op/ must be semistable,and the result now follows from Theorem 5.17.



Theorem 5.28 There is a canonical isomorphism h W P totM .X; 1; n/! .M ss.X//nC1 .

Proof Define h0 W P totM .X; 1; n/!M ss.X/ by

h0.ŒE; `q1; p̀1

; : : : ; p̀n�/D ŒE�:

For i D 1; : : : ; n, choose a point ei 2 X such that q1 C pi D 2ei and definehi W P tot

M .X; 1; n/!M ss.X/ by

hi .ŒE; `q1; p̀1

; : : : ; p̀n�/D ŒH.E; `q1

; p̀i/˝O.ei /�:

Then h WD .h0; h1; : : : ; hn/ is an isomorphism by Theorem 3.20, Corollary 5.25 andLemma 5.26.

For nD 1, the isomorphism h W P totM .X; 1; n/! .M ss.X//nC1 appears to be closely

related to an isomorphism M ss.X; 2/! .CP1/2 defined in [28], and our definitionof h was motivated by this isomorphism.

5.4 Embedding PM .X; m; n/ ! M s.X; m C n/

We will now describe a canonical open embedding of the space PM .X;m; n/ into thespace of stable parabolic bundles M s.X;mCn/. We first need two lemmas:

Lemma 5.29 Let .E; p̀1; : : : ; p̀n

/ be a parabolic bundle over an elliptic curve Xsuch that E is semistable. If the lines p̀1

; : : : ; p̀nare bad in the same direction then

H.E; p̀1; : : : ; p̀n

/ has instability degree n.

Proof Up to tensoring with a line bundle, the bundle E has one of three forms:

(1) (E D O˚O) Since p̀1; : : : ; p̀n

are bad in the same direction, we have that

p̀1D � � � D p̀n

under a global trivialization of E in which all the fibers are identifiedwith C2 . A sequence of Hecke modifications with p̀1

D � � � D p̀nis given by

O˚O ˛1

p1 �O˚O.�p1/

˛2

p2 � � � �

˛n

pn �O˚O.�p1� � � � �pn/:

Here O˚O ˛1

p1 �O˚O.�p1/ is a Hecke modification corresponding to p̀1

, and fori D 2; : : : ; n we define

˛i D

�1 0

0 fi

�;

where fi is the unique (up to rescaling by a constant) morphism from O.�p1�� pi /to O.�p1�� pi�1/. Thus H.O˚O; p̀1

; : : : ; p̀n/DO˚O.�p1�� pn/ has

instability degree n.


584 David Boozer

(2) (E D F2 ) Then p̀iDOpi

for i D 1; : : : ; n. A sequence of Hecke modificationswith p̀i

DOpifor i D 1; : : : ; n is given by

F2˛1

p1 �O˚O.�p1/

˛2

p2 � � � �

˛n

pn �O˚O.�p1� � � � �pn/;

where F2˛1

p1 �O˚O.�p1/ is a Hecke modification corresponding to p̀1

DOp1and

˛i is as above for i D 2; : : : ; n. Thus H.F2;Op1; : : : ;Opn

/DO˚O.�p1�� pn/has instability degree n.

(3) (E D L˚L�1 for a degree 0 line bundle L such that L2 ¤ O) Then either

p̀iD Lpi

for i D 1; : : : ; n or p̀iD .L�1/pi

for i D 1; : : : ; n. A sequence of Heckemodifications with p̀i

D Lpifor i D 1; : : : ; n is given by

L˚L�1˛1

p1 � L˚ .L�1˝O.�p1//

˛2

p2 � � � �

˛n

pn � L˚ .L�1˝O.�p1� � � � �pn//;

where˛i D

�1 0

0 1˝fi

�and fi is as above. Thus H.L˚L�1; Lp1

; : : : ; Lpn/DL˚.L�1˝O.�p1�� pn//

has instability degree n. We can write down a similar sequence of Hecke modificationsto show that H.L˚L�1; .L�1/p1

; : : : ; .L�1/pn/D .L˝O.�p1� � � ��pn//˚L�1

has instability degree n.

Using Lemma 5.29 in place of Lemma 4.16, the proofs of Lemma 4.17 and Theorem 4.19for rational curves carry over to the case of elliptic curves. We thus obtain:

Lemma 5.30 Let .E; p̀1; : : : ; p̀n

/ be a parabolic bundle over an elliptic curve Xsuch that E is semistable. If H.E; p̀1

; : : : ; p̀n/ is semistable, .E; p̀1

; : : : ; p̀n/ is

semistable.

Theorem 5.31 There is a canonical open embedding PM .X;m; n/!M s.X;mCn/.

5.5 Examples

Here we compute the space PM .X; 1; n/ for nD0; 1; 2. We first make some definitions:

Definition 5.32 The Abel–Jacobi isomorphism X!Jac.X/ is given by p 7! ŒO.p�e/�for a choice of basepoint e 2X.

Definition 5.33 We define a map � W Jac.X/!M ss.X/, ŒL� 7! ŒL˚L�1�.



Note that � is surjective and �.L/D �.L�1/, so � W Jac.X/ŠX!M ss.X/ŠCP1

is a two-to-one branched cover with four branch points ŒLi ˚Li � corresponding to thefour 2–torsion line bundles Li .

Definition 5.34 Given a degree 0 divisor D on an elliptic curve X, define the trans-lation map �D W Jac.X/! Jac.X/, ŒL� 7! ŒL˝O.D/�.

5.5.1 Calculate PM .X; 1; 0/ We have that

PM .X; 1; 0/D P totM .X; 1; 0/DM

s.X; 1/DM ss.X/DCP1:

Note that the embedding PM .X; 1; 0/! M s.X; 1/ defined in Theorem 5.31 is anisomorphism.

5.5.2 Calculate PM .X; 1; 1/

Theorem 5.35 The map g W Jac.X/ ! .M ss.X//2 , g D .�; � ı �p1�e1/, is injec-

tive and has image the complement of h.PM .X; 1; 1//, where h W P totM .X; 1; 1/ !

.M ss.X//2 is the isomorphism described in Theorem 5.28.

Proof First we show that g has image the complement of h.PM .X; 1// in .M ss.X//2

Take a point ŒE; `q1; p̀1

�2P totM .X; 1; 1/. From Theorems 5.21, 5.22 and 5.24, it follows

that H.E; p̀1/ D G2.p1/˝O.�p1/ is stable if p̀1

is a good line, and H.E; p̀1/

is unstable if p̀1is a bad line. So the complement of PM .X; 1; 1/ in P tot

M .X; 1; 1/

consists of isomorphism classes ŒE; `q1; p̀1

� such that p̀1is a bad line, and is thus

given by the union of the sets

S1 D fŒL˚L�1; `q1

; Lp1� j ŒL� 2 Jac.X/; L2 ¤Og;

S2 D fŒL˚L�1; `q1

; .L�1/p1� j ŒL� 2 Jac.X/; L2 ¤Og;

S3 D fŒF2˝Li ; `q1; .Li /p1

� j i D 1; 2; 3; 4g;

where in each case `q1is a good line. From Lemma 5.27, it follows that the complement

of h.PM .X; 1; 1// in .M ss.X//2 is given by the union of the sets

h.S1/D˚��.ŒL�/; .� ı �p1�e1

/.ŒL�/�j ŒL� 2 Jac.X/; L2 ¤O

;

h.S2/D˚��.ŒL�/; .� ı �e1�p1

/.ŒL�/�j ŒL� 2 Jac.X/; L2 ¤O

;

h.S3/D˚��.ŒLi �/; .� ı �p1�e1

/.ŒLi �/�j i D 1; 2; 3; 4

:

Note that ��.ŒL�/; .� ı �e1�p1

/.ŒL�/�D��.ŒL�1�/; .� ı �p1�e1

/.ŒL�1�/�;


586 David Boozer

so h.S1/D h.S2/, and we have that

h.S1/[ h.S2/[ h.S3/D˚��.ŒL�/; .� ı �p1�e1

/.ŒL�/�j ŒL� 2 Jac.X/

D img:

So the image of g is the complement of h.PM .X; 1; 1// in .M ss.X//2 .

Next we show that g is injective. If g.L/D g.L0/, then projection onto the first factorof .M ss.X//2 gives �.L/ D �.L0/, hence either L0 D L or L0 D L�1 . SupposeL0 D L�1 . Then projection onto the second factor of .M ss.X//2 gives

�.L˝O.p1� e1//D �.L�1˝O.p1� e1//;

hence either L˝O.p1�e1/DL�1˝O.p1�e1/ or L˝O.p1�e1/DL˝O.e1�p1/.The first case implies LD L�1 . The second case implies 2p1 D 2e1 , but we chosee1 such that p1C q1 D 2e1 , hence p1 D q1 , a contradiction. Thus L0 D L, so g isinjective.

If we use the Abel–Jacobi isomorphism to identify X and Jac.X/, the (canonical)isomorphism h W P tot

M .X; 1; 1/! .M ss.X//2 to identify P totM .X; 1; 1/ and .M ss.X//2 ,

and the (noncanonical) isomorphism M ss.X/ŠCP1 to identify M ss.X/ and CP1 ,we find that

PM .X; 1; 1/D .CP1/2�g.X/:

Remark 5.36 Using results from the proof of Theorem 5.35, it is straightforward toshow that

M ss.X; 2/D P totM .X; 1; 1/D .CP1/2; M s.X; 2/D PM .X; 1; 1/D .CP1/2�g.X/:

These calculations reproduce the results of [28] for M ss.X; 2/ and M s.X; 2/.

5.5.3 Calculate PM .X; 1; 2/ The same method that we used to prove Theorem 5.35can be used to calculate PM .X; 1; 2/:

Theorem 5.37 The map f W Jac.X/! .M ss.X//3 , f D .�; �ı�p1�e1; �ı�p2�e2

/, isinjective and has image the complement of h.PM .X; 1; 2//, where h W P tot

M .X; 1; 2/!

.M ss.X//3 is the isomorphism described in Theorem 5.28.

If we use the Abel–Jacobi isomorphism to identify X and Jac.X/, the (canonical)isomorphism h W P tot

M .X; 1; 2/! .M ss.X//3 to identify P totM .X; 1; 2/ and .M ss.X//3 ,

and the (noncanonical) isomorphism M ss.X/ŠCP1 to identify M ss.X/ and CP1 ,we find that

PM .X; 1; 2/D .CP1/3�f .X/:



6 Possible applications to topology

Here we briefly outline some possible applications of our results to topology. Wehave proposed complex manifolds PM .X; 1; 2r/ and PM .X; 3; 2r/ as candidates fora space Y.T 2; 2r/ that generalizes the Seidel–Smith space Y.S2; 2r/ and that couldpotentially be used to construct symplectic Khovanov homology for lens spaces. Thefollowing tasks remain to be done to complete the construction:

(1) We need to define a suitable symplectic form on PM .X;m; 2r/. One possibilityis to pull back the canonical symplectic form on M s.X; 2rCm/ using the openembedding PM .X;m; 2r/!M s.X; 2r Cm/.

(2) We need to find a suitable action of the mapping class group MCG2r.T 2/ onPM .X;m; 2r/ that is defined up to Hamiltonian isotopy. Such an action mightbe obtained via symplectic monodromy by viewing PM .X;m; 2r/ as the fiberof a larger space that fibers over the moduli space of genus 1 curves with markedpoints. Such an approach would be analogous to the way Seidel and Smithobtain an action of the braid group on the Seidel–Smith space via monodromyaround loops in the configuration space [23], and similar methods are used todefine mapping class group actions for constructing Reshetikhin–Turaev–Witteninvariants [3].

(3) We need to define suitable Lagrangians Lr in PM .X;m; 2r/ corresponding tor unknotted arcs in a solid torus. For PM .X; 1; 2r/ we would expect Lr to behomeomorphic to S1 � .S2/r , and for PM .X; 3; 2r/ we would expect Lr tobe homeomorphic to S3 � .S2/r . Perhaps such Lagrangians can be constructedin a manner analogous to Seidel and Smith by viewing PM .X;m; 2r/ as thefiber of a larger space that fibers over the configuration space Conf2r.X/ of 2runordered points in X and looking for vanishing cycles as points are successivelybrought together in pairs.

(4) We need to prove that the Lagrangian Floer homology of a knot K in a lensspace Y is invariant under different Heegaard splittings of .Y;K/ into solid tori.

(5) We need to verify that our construction of symplectic Khovanov homologyreproduces ordinary Khovanov homology for the case of knots in S3 .

Several of our results appear to be related to a possible connection between Khovanovhomology and symplectic instanton homology. Roughly speaking, symplectic instantonhomology is defined as follows. Given a knot K in a 3–manifold Y , one Heegaard-splits .Y;K/ along a Heegaard surface † to obtain handlebodies U1 and U2 . Each


588 David Boozer

Figure 1: Left to right: (a) The graph ‚; (b) the graph � in B3 ; (c) the graphDp � S

3 for p D 1; (d) the graph � in S1 �D2 .

handlebody Ui contains a portion of the knot Ai WD Ui \K consisting of r arcs thatpairwise connect points p1; : : : ; p2r in †. To the marked surface .†; p1; : : : ; p2r/one associates a character variety R.†; 2r/, which has the structure of a symplecticmanifold, and to the handlebody pairs .Ui ; Ai / one associates Lagrangians Li �R.†; 2r/. The symplectic instanton homology of .Y;K/ is then defined to be theLagrangian Floer homology of the pair of Lagrangians .L1; L2/.

In fact, there are several technical difficulties that must be overcome in order to geta well-defined homology theory. For example, one needs to introduce a framingin order to eliminate singularities in the character variety R.†; 2r/. One way tointroduce a framing is by replacing the knot K with K [‚, where ‚ is the thetagraph shown in Figure 1(a); this approach is described in [10]. We Heegaard-split.Y;K [‚/ along a Heegaard surface † that is chosen to transversely intersect eachedge ei of the theta graph in a single point qi . The marked Heegaard surface is now.†; q1; q2; q3; p1; : : : ; p2r/, corresponding to the character variety R.†; 2rC3/, andthe handlebody pairs are now .Ui ; Ai [ �i /, where �i is the epsilon graph shown inFigure 1(b). The character variety R.†; 2r C 3/ has the structure of a symplecticmanifold that is symplectomorphic to the moduli space of stable parabolic bundlesM s.C; 2r C 3/, where C is any complex curve homeomorphic to †. (The spaceM s.C; 2r C 3/ has a canonical symplectic form.)

Symplectic instanton homology can be viewed as a symplectic replacement for singularinstanton homology, a knot homology theory defined using gauge theory, and the twotheories are conjectured to be isomorphic. This is an example of an Atiyah–Floer con-jecture; such conjectures broadly relate Floer-theoretic invariants defined using gaugetheory to corresponding invariants defined using symplectic topology. Kronheimer andMrowka constructed a spectral sequence from Khovanov homology to singular instantonhomology [17], and the embedding PM .CP1; 2r/! M s.CP1; 2r C 3/ describedin Section 4.4 suggests that it may be possible to construct an analogous spectral



sequence from symplectic Khovanov homology to symplectic instanton homology.(This idea for constructing a spectral sequence was suggested to the author by IvanSmith and Chris Woodward.) If so, perhaps the fact that we have an embeddingPM .X; 3; 2r/ ! M s.X; 2r C 3/, as described in Section 5.4, is evidence that thecorrect generalization of the Seidel–Smith space is PM .X; 3; 2r/. Indeed, a calculationof the Lagrangian intersection L1 \L2 in the traceless character variety R.T 2; 3/for .S3; ‚/ yields a single point, a space whose cohomology is the correct Khovanovhomology for the empty knot, and calculations of the Lagrangian intersections in thetraceless character variety R.T 2; 5/ for .S3; unknot[‚/ and .S3; trefoil[‚/ yieldS2 and RP3qS2 , spaces whose cohomology gives the correct Khovanov homology forthe unknot and trefoil. Based on these speculations, we make the following conjectures:

Conjecture 6.1 The space PM .C; 3; 2r/ is the correct generalization of the Seidel–Smith space Y.S2; 2r/ to a curve C of arbitrary genus.

Conjecture 6.2 Given a curve C of arbitrary genus , there is a canonical open embed-ding PM .C;m; n/!M s.C;mCn/.

On the other hand, perhaps the embedding PM .X; 1; 2r/!M s.X; 2rC1/ described inSection 5.4 is related to a spectral sequence from a Khovanov-like knot homology theoryto symplectic instanton homology defined with a novel framing. Rather than using atheta graph, perhaps for the lens space L.p; q/ one could introduce a framing specificto that lens space by using a p–linked dumbbell graph Dp , as shown in Figure 1(c)for the case p D 1. There is a unique edge e1 of the dumbbell graph that connects thetwo vertices, and one can choose a Heegaard surface † that transversely intersects e1in a single point q1 . The marked Heegaard surface is now .†; q1; p1; : : : ; p2r/,corresponding to the character variety R.†; 2rC1/, and the handlebody pairs .Ui ; Ai /are now .Ui ; Ai [ �i /, where �i is the sigma graph shown in Figure 1(d). Thecharacter variety R.†; 2r C 1/ has the structure of a symplectic manifold that issymplectomorphic to the moduli space of stable parabolic bundles M s.X; 2r C 1/,which is the codomain of the embedding PM .X; 1; 2r/!M s.X; 2r C 1/.

Appendix A Vector bundles

Here we briefly review some results on holomorphic vector bundles and their modulispaces that we will use throughout the paper. Some useful references on vector bundlesare [18; 22; 26; 27].


590 David Boozer

Definition A.1 The slope of a holomorphic vector bundle E over a curve C isslopeE WD .degE/=.rankE/ 2Q.

Definition A.2 A holomorphic vector bundle E over a curve C is stable if slopeF <slopeE for any proper subbundle F � E, semistable if slopeF � slopeE for anyproper subbundle F � E, strictly semistable if it is semistable but not stable, andunstable if there is a proper subbundle F �E such that slopeF > slopeE.

If E is a stable vector bundle, then Aut.E/DC� consists only of trivial automorphismsthat scale the fibers by a constant factor.

Definition A.3 Given a semistable vector bundle E, a Jordan–Hölder filtration of Eis a filtration

F0 D 0� F1 � F2 � � � � � Fn DE

of E by subbundles Fi �E for iD0; : : : ; n such that the composition factors Fi=Fi�1are stable and slopeFi=Fi�1 D slopeE for i D 1; : : : ; n.

Every semistable vector bundle E admits a Jordan–Hölder filtration. The filtration isnot unique, but the composition factors Fi=Fi�1 for i D 1; : : : ; n are independent (upto permutation) of the choice of filtration.

Definition A.4 Given a semistable holomorphic vector bundle E over a curve C, theassociated graded vector bundle grE is defined to be

grE DnMiD1

Fi=Fi�1;

where F0 D 0� F1 � � � � � Fn DE is a Jordan–Hölder filtration of E.

The bundle grE is independent (up to isomorphism) of the choice of filtration, andslope.grE/D slopeE.

Definition A.5 Two semistable vector bundles are said to be S –equivalent if theirassociated graded bundles are isomorphic.

Example A.6 In Section 5.1 we define a strictly semistable rank 2 vector bundle F2and a stable rank 2 vector bundle G2.p/ over an elliptic curve X. A Jordan–Hölderfiltration of F2 is O � F2 , and the associated graded bundle is grF2 D O˚O. Itfollows that F2 and O˚O are S –equivalent. A Jordan–Hölder filtration of G2.p/ isjust G2.p/, and the associated graded bundle is grG2.p/DG2.p/.



Isomorphic bundles are S –equivalent. For rational curves, S –equivalent bundles areisomorphic, but this is not true in general. For example, on an elliptic curve the bundlesF2 and O˚O are S –equivalent but not isomorphic.

Definition A.7 We define M ss.C / (respectively M s.C /) to be the moduli space ofsemistable (respectively stable) rank 2 holomorphic vector bundles over curve C withtrivial determinant bundle, mod S –equivalence. This space is defined in [25]; seealso [20].

Remark A.8 An alternative way of interpreting M ss.C / is as the space of flat SU.2/–connections on a trivial rank 2 complex vector bundle E! C, mod gauge transfor-mations. Yet another way of interpreting the space M ss.C / is as the character varietyR.C/ of conjugacy classes of group homomorphisms �1.C /! SU.2/. We will notuse these interpretations here.

The moduli space M s.C / has the structure of a complex manifold of dimension3.g� 1/, where g is the genus of the curve C. The space M s.C / carries a canonicalsymplectic form, which is obtained by interpreting M s.C / as a Hamiltonian reductionof a space of SU.2/–connections.

Example A.9 For rational curves, the bundle O˚O is the unique semistable rank 2bundle with trivial determinant bundle, and there are no stable rank 2 bundles, so

M ss.CP1/D fptg D fŒO˚O�g; M s.CP1/D¿:

Example A.10 For an elliptic curve X, semistable rank 2 bundles with trivial deter-minant bundle have the form L˚L�1 , where L is a degree 0 line bundle, or F2˝Li ,where Li for i D 1; : : : ; 4 are the four 2–torsion line bundles. The bundles Li ˚Liand F2˝Li are S –equivalent. The bundles L˚L�1 and L�1˚L are isomorphic,hence S –equivalent. There are no stable rank 2 bundles with trivial determinant bundle.As shown in [27], we have that

M ss.X/D fŒL˚L�1� j ŒL� 2 Jac.X/g DCP1; M s.X/D¿:

Appendix B Parabolic bundles

Here we briefly review some results on parabolic bundles and their moduli spacesthat we will use throughout the paper. Some useful references on parabolic bundlesare [19; 21].


592 David Boozer

B.1 Definition of a parabolic bundle

The concept of a parabolic bundle was introduced in [19]:

Definition B.1 A parabolic bundle of rank r on a curve C consists of following data:

(1) A rank r holomorphic vector bundle �E WE! C.

(2) Distinct marked points .p1; : : : ; pn/ 2 C n .

(3) For each marked point pi , a flag of vector spaces Ejpiin the fiber Epi

D��1E .pi /

over the point pi :

E0piD 0�E1pi

�E2pi� � � � �Esipi

DEpi:

(4) For each marked point pi , a strictly decreasing list of weights �jpi2R:

�1pi> �2pi

> � � �> �sipi:

We refer the data of the marked points, the flags and the weights as a parabolic structureon E. We refer to the data of just the marked points and flags, without the weights,as a quasiparabolic structure on E. We define the multiplicity of the weight �jpi

tobe mjpi

WD dim.Ejpi/� dim.Ej�1pi

/. The definition of a parabolic bundle given in [19]differs slightly from our definition, in that the marked points are unordered and theweights are required to lie in the range Œ0; 1/.

Definition B.2 Two parabolic bundles with underlying vector bundles E and F areisomorphic if the marked points and weights for the two bundles are the same, and thereis a bundle isomorphism ˛ WE! F that carries each flag of E to the correspondingflag of F ; that is, ˛.Ejpi

/D Fjpi

for j D 1; : : : ; si and i D 1; : : : ; n.

Definition B.3 The rank of a parabolic bundle is the rank of its underlying vectorbundle.

Definition B.4 The parabolic degree and parabolic slope of a parabolic bundle Ewith underlying vector bundle E are defined to be

pdeg E D degECnXiD1

siXjD1

mjpi�jpi2R; pslope E D

pdeg Erank E

2Q:

We will not need the full generality of the concept of a parabolic bundle; rather, wewill consider only parabolic line bundles and rank 2 parabolic bundles of a certainrestricted form.



First we consider parabolic line bundles. For such bundles there is no flag data, sothe parabolic structure is specified by a list of marked points p1; : : : ; pn and a list ofweights �1p1

; : : : ; �1pn. We fix a parameter � > 0 and restrict to the case �1pi

2 f˙�g

for i D 1; : : : ; n. A parabolic line bundle of this form thus consists of the data.L; �p1

; : : : ; �pn/, where �L W L! C is a holomorphic line bundle and �pi

2 f˙1g.The parabolic degree and parabolic slope of a parabolic line bundle .L; �p1

; : : : ; �pn/

are given by

pdeg.L; �p1; : : : ; �pn

/D pslope.L; �p1; : : : ; �pn

/D degLC�nXiD1

�pi:

Next we consider rank 2 parabolic bundles. We fix a parameter � > 0 and restrictto the case si D 2, m1pi

D m2piD 1, and �1pi

D ��2piD � for i D 1; : : : ; n. A

rank 2 parabolic bundle of this form thus consists of the data .E; p̀1; : : : ; p̀n

/, where�E WE ! C is a rank 2 holomorphic vector bundle and p̀i

2 P .Epi/ is a line in

the fiber EpiD ��1E .pi / over the point pi for i D 1; : : : ; n. The parabolic slope and

parabolic degree of a rank 2 parabolic bundle .E; p̀1; : : : ; p̀n

/ are given by

pdeg.E; p̀1; : : : ; p̀n

/D degE; pslope.E; p̀1; : : : ; p̀n

/D slopeE:

B.2 Stable, semistable and unstable parabolic bundles

Consider a rank 2 parabolic bundle .E; p̀1; : : : ; p̀n

/ and a line subbundle L � E.There are induced parabolic structures on the line bundles L and E=L given by.L; �p1

; : : : ; �pn/ and .E=L;��p1

; : : : ;��pn/, where

�piD

�C1 if Lpi

D p̀i,

�1 if Lpi¤ p̀i

.

Definition B.5 Given a rank 2 parabolic bundle .E; p̀1; : : : ; p̀n

/ and a line subbundleL�E, we say that the induced parabolic bundle .L; �p1

; : : : ; �pn/ is a parabolic sub-

bundle of .E; p̀1; : : : ; p̀n

/ and the induced parabolic bundle .E=L;��p1; : : : ;��pn

/

is a parabolic quotient bundle of .E; p̀1; : : : ; p̀n

/.

Definition B.6 A rank 2 parabolic bundle .E; p̀1; : : : ; p̀n

/ is said to be decompos-able if there exists a decomposition E D L˚ L0 for line bundles L and L0 suchthat p̀i

2 fLpi; L0pig for i D 1; : : : ; n. For a rank 2 decomposable parabolic bundle

.E; p̀1; : : : ; p̀n

/ we write

.E; p̀1; : : : ; p̀n

/D .L; �p1; : : : ; �pn

/˚ .L0; � 0p1; : : : ; � 0pn

/;


594 David Boozer

where .L; �p1; : : : ; �pn

/ and .L0; � 0p1; : : : ; � 0pn

/ are the induced parabolic structureson L and L0.

Definition B.7 A rank 2 parabolic bundle is stable if its parabolic slope is strictlygreater than the parabolic slope of any of its proper parabolic subbundles, semistable ifits parabolic slope is greater than or equal than the parabolic slope of any of its properparabolic subbundles, strictly semistable if it is semistable but not stable, and unstableif it has a proper parabolic subbundle of strictly greater slope.

If E is a stable parabolic bundle, then Aut.E/DC� consists only of trivial automor-phisms that scale the fibers of the underlying vector bundle by a constant factor.

Theorem B.8 If the rank 2 parabolic bundle .E; p̀1; : : : ; p̀n

/ is semistable and� < 1=2n, then E is semistable.

Proof We will prove the contrapositive, so assume that E is unstable. Then there is aline subbundle L�E such that slopeL > slopeE. Consider the parabolic structure.L; �p1

; : : : ; �pn/ induced on L by .E; p̀1

; : : : ; p̀n/. We have that

pslope.L; �p1; : : : ; �pn

/� pslope.E; p̀1; : : : ; p̀n

/D slopeLC�nXiD1

�pi� slopeE:

Since slopeL is an integer, slopeE is an integer or half-integer and slopeL> slopeE,it follows that slopeL� slopeE � 1

2. From the assumption that � < 1=2n, it follows

thatpslope.L; �p1

; : : : ; �pn/� pslope.E; p̀1

; : : : ; p̀n/� 1

2�n� > 0;

so .E; p̀1; : : : ; p̀n

/ is unstable.

Throughout this paper we will always assume � � 1, by which we mean that �is always chosen to be sufficiently small that Theorem B.8 holds under whatevercircumstances we are considering.

Consider a rank 2 vector bundle E over a curve C. If E is unstable, then Theorem B.8implies that the parabolic bundle .E; p̀1

; : : : ; p̀n/ is unstable. If E is semistable, then

the stability of the parabolic bundle .E; p̀1; : : : ; p̀n

/ can be characterized as follows:

Theorem B.9 Consider a rank 2 parabolic bundle of the form .E; p̀1; : : : ; p̀n

/ withE semistable. Let m be the maximum number of lines that are bad in the same



direction. Such a parabolic bundle is stable if and only if m< 12n, semistable if and

only if m � 12n, and unstable if and only if m > 1

2n. In particular, if n is odd then

stability and semistability are equivalent.

Example B.10 As a special case of Theorem B.9, consider parabolic bundles

.E; p̀1; : : : ; p̀n

/

over CP1 with underlying vector bundle EDO˚O. We can globally trivialize E andidentify all the fibers with C2 . All lines of E are bad, and lines are bad in the samedirection if and only if they are equal under the global trivialization. Let m denote themaximum number of lines p̀i

equal to any given line in CP1 . From Theorem B.9,we find that .E; p̀1

; : : : ; p̀n/ is stable if m< 1

2n, semistable if m� 1

2n, and unstable

if m > 12n. For example, .E; p̀1

/ is unstable, .E; p̀1; p̀2

/ is strictly semistable ifthe lines are distinct and unstable otherwise, and .E; p̀1

; p̀2; p̀3

/ is stable if the linesare distinct and unstable otherwise.

B.3 S –equivalent semistable parabolic bundles

There is a Jordan–Hölder theorem for parabolic bundles, which asserts that anysemistable parabolic bundle of parabolic degree 0 has a filtration in which quotients ofsuccessive parabolic bundles (composition factors) in the filtration are stable withparabolic slope 0 (see [19, Remark 1.16]). The filtration is not unique, but thecomposition factors are unique up to permutation. It follows that one can definean associated graded bundle of a semistable parabolic bundle of parabolic degree 0that is unique up to isomorphism.

We will need the concept of an associated graded parabolic bundle only for the case ofsemistable rank 2 parabolic bundles. If such a parabolic bundle E is stable, then itsassociated graded parabolic bundle gr E is just E. Now consider a strictly semistableparabolic bundle E D .E; p̀1

; : : : ; p̀n/. Under our standard assumption that �� 1,

it follows from Theorem B.8 that E is semistable. The associated graded parabolicbundle gr E is given by

gr.E; p̀1; : : : ; p̀n

/D .L; �p1; : : : ; �pn

/˚ .E=L;��p1; : : : ;��pn

/;

where L�E is a line subbundle such that slopeLD slopeE, and .L; �p1; : : : ; �pn

/

and .E=L;��p1; : : : ;��pn

/ are the induced parabolic structures on L and E=L. Notethat

pslope.gr.E; p̀1; : : : ; p̀n

//D pslope.E; p̀1; : : : ; p̀n

/D slopeE:


596 David Boozer

Definition B.11 We say that two semistable rank 2 parabolic bundles are S –equivalentif their associated graded bundles are isomorphic.

Isomorphic parabolic bundles are S –equivalent. Here we give an example to show thatthe converse does not always hold:

Example B.12 Consider parabolic bundles over CP1 with underlying vector bundleE DO˚O. We can globally trivialize E and identify all the fibers with C2 . Let A,B and C be distinct lines in CP1 , and consider the two strictly semistable parabolicbundles

E WD .E; p̀1D A; p̀2

D A; p̀3D B; p̀4

D C/;

E 0 WD .E; `0p1D B; `0p2

D C; `0p3D A; `0p4

D A/:

The bundles E and E 0 are not isomorphic but are S –equivalent, since the associatedgraded bundles of both bundles are isomorphic to L˚L0, where

L WD .O; �p1D 1; �p2

D 1; �p3D�1; �p4

D�1/;

L0 WD .O; �p1D�1; �p2

D�1; �p3D 1; �p4

D 1/:

B.4 Moduli spaces of rank 2 parabolic bundles

Definition B.13 We define M ss.C;n/ (respectively M s.C;n/), to be the moduli spaceof semistable (respectively stable) rank 2 parabolic bundles of the form .E; p̀1

; : : : ; p̀n/

such that E has trivial determinant bundle, mod S –equivalence. In particular,

M ss.C; 0/DM ss.C / and M s.C; 0/DM s.C /:

As always, we assume that �� 1. This space is defined in [19]; see also [4].

Remark B.14 An alternative way of interpreting M ss.C; n/ is as the space of flatSU.2/–connections on a trivial rank 2 complex vector bundle E! C �fp1; : : : ; png,where the holonomy around each puncture point pi is conjugate to diag.e2�i�; e�2�i�/,mod SU.2/ gauge transformations. Yet another way of interpreting the space M ss.C; n/

is as the character variety R.C; n/ of conjugacy classes of group homomorphisms�1.C �fp1; : : : ; png/! SU.2/ that take loops around the marked points to matricesconjugate to diag.e2�i�; e�2�i�/. Note that � D 1

4corresponds to the traceless

character variety. We will not use these interpretations here.



For rational �, the moduli space M s.C; n/ has the structure of a complex manifold ofdimension 3.g�1/Cn, where g is the genus of the curve C, and M s.C; n/ is compactfor n odd. The space M s.C; n/ carries a canonical symplectic form, which is obtainedby viewing M s.C; n/ as a Hamiltonian reduction of a space of SU.2/–connectionswith prescribed singularities.

Example B.15 Let C D CP1 be a rational curve. If we fix n � 3, then all rank 2parabolic bundles of the form .O˚O; p̀1

; : : : ; p̀n/ for which all the lines are distinct

are isomorphic. From this fact, together with the results of Example B.10, we find that

M ss.CP1; 0/D fptg; M s.CP1; 0/D¿;

M ss.CP1; 1/D¿; M s.CP1; 1/D¿;

M ss.CP1; 2/D fptg; M s.CP1; 2/D¿;

M ss.CP1; 3/D fptg; M s.CP1; 3/D fptg:

Using the cross-ratio and considerations of S –equivalence as described in Example B.12,one can show

M ss.CP1; 4/DCP1; M s.CP1; 4/DCP1�f3 pointsg:

Example B.16 Let X be an elliptic curve. From Corollary 5.25, Example A.10 andTheorem B.9, we have that

M ss.X; 0/DCP1; M s.X; 0/D¿; M ss.X; 1/DCP1; M s.X; 1/DCP1:

In [28] it is shown that

M ss.X; 2/D .CP1/2; M s.X; 2/D .CP1/2�g.X/;

where g WX ! .CP1/2 is a holomorphic embedding. We also derive this result inSection 5.5.2.

Remark B.17 Throughout this paper we assume �� 1, but for some applicationsone wants to take �D 1

4in order to interpret M ss.C; n/ as a traceless character variety,

as described in Remark B.14. In general, M ss.C; n/ depends on �; for example, for0� n� 4 the space M ss.CP1; n/ is the same for �� 1

4and �D 1

4, but, as shown

in [23], for nD 5 we have

M ss.CP1; 5/D

�CP2 # 4CP2 for �� 1,CP2 # 5CP2 for �D 1

4.

The dependence of M ss.C; n/ on � is discussed in [6].


598 David Boozer

Acknowledgments

The author would like to express his gratitude towards Ciprian Manolescu for suggestingthe problem considered in this paper and for providing invaluable guidance, JoelKamnitzer for helpful discussions, Burt Totaro for helpful advice and for offeringextensive comments on an earlier version of this paper that led to significant changesin the current version, and Chris Woodward for sharing his personal notes on theWoodward embedding described in Section 4.6.2. The author was partially supportedby NSF grant number DMS-1708320.

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Department of Mathematics, Princeton UniversityPrinceton, NJ, United States

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Received: 6 July 2018 Revised: 28 April 2020

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