Variation of Geometric Invariant Theory and Derived Categoriesfavero/slides/VGIT.pdf · 2020-05-08 · Variation of Geometric Invariant Theory and Derived Categories David Favero

Variation of Geometric Invariant Theory andDerived Categories

David Favero

University of Vienna

June 6, 2012

David Favero VGIT and Derived Categories

Attributions

Based on joint work with M. Ballard (U. Wisconsin) and LudmilKatzarkov (U. Miami and U. Vienna).

Available at http://arxiv.org/abs/1203.6643.


Outline

1 Motivating Example

2 Background on GIT

3 General results

4 Landau-Ginzburg models and factorizations

5 RG-flow and a theorem of Orlov


Motivating Example

Outline


2 Background on GIT

3 General results




Motivating Example

Weighted Projective Stacks

Consider k[x0, x1, x2] with the Gm-action with weights (1, 1, n). Wedefine P(1 : 1 : n) as the smooth global quotient Deligne-Mumfordstack, [(Spec k[x0, x1, x2]\0)/Gm]. Characters of Gm, λ 7→ λi, giveline bundles, O(i), and a tilting object, T , is given by,

T :=n+2⊕i=0

O(i)

Quiver for P(1 : 1 : 4):

• • • • • •x0

x1

x0

x1

x0

x1

x0

x1

x0

x1

x2

x2


Motivating Example

Hirzebruch Surfaces

Consider the total space of OP1 ⊕OP1(−n) with the Gm-action givenby dilating the fibers. The Hirzebruch surface, Fn, is defined as theprojective bundle, P(OP1 ⊕OP1(−n)), which represents the smoothglobal quotient stack, [tot(OP1 ⊕OP1(−n))\zero section/Gm]. Atilting object, T , is given by,T := O ⊕ π∗O(1)⊕Oπ(1)⊕ π∗O(1)⊗Oπ(1).

• •

••

x2 x2

x0

x1

x1

x0

x30

x31

x20x1

x0x21


Motivating Example

Comparing

Quiver for P(1 : 1 : 4):

• • • • • •x0

x1

x0

x1

x0

x1

x0

x1

x0

x1

x2

x2

Quiver for Fn:

• •

••

x2 x2

x0

x1

x1

x0

x30

x31

x20x1

x0x21


Motivating Example

Semi-orthogonal decompositions

DefinitionA semi-orthogonal decomposition of a triangulated category, T , is asequence of full triangulated subcategories, A1, . . . ,Am, in T suchthat Ai ⊂ A⊥j for i < j and, for every object T ∈ T , there exists adiagram:

0 Tm−1 · · · T2 T1 T

Am A2 A1

|||

where all triangles are distinguished and Ak ∈ Ak. We denote asemi-orthogonal decomposition by 〈A1, . . . ,Am〉.


Motivating Example

Hirzebruch surfaces

• If n < 2, there is a semi-orthogonal decomposition

Db(coh Fn) = 〈E1, . . . ,E2−n,Db(coh P(1, 1, n))〉

with Ei exceptional objects.• If n = 2, we have an equivalence

Db(coh Fn) = Db(coh P(1, 1, 2)).

• If n > 2, there is a semi-orthogonal decomposition

Db(coh P(1, 1, n)) = 〈E1, . . . ,En−2,Db(coh Fn)〉

with Ei exceptional objects.


Background on GIT

Outline


2 Background on GIT

3 General results




Background on GIT

Reminder on VGIT

X is a smooth quasi-projective variety over an algebraicallyclosed field, k of characteristic zero,

G is a linearly reductive algebraic group acting on X,

L is a G-equivariant ample line bundle on X,

The semi-stable locus is an open subset,

Xss(L) := {x ∈ X | ∃f ∈ H0(X,Ln)G with n ≥ 0, f (x) 6= 0, and Xf affine}

For us, the GIT quotient corresponding to this data is the globalquotient stack [Xss(L)/G]. We can vary the G-equivariant structure onL by choosing characters, χ, in the dual group, G := Hom(G,Gm).We denote the GIT quotient corresponding to this linearization byX//L(χ).


Background on GIT

Reminder on VGIT

The unstable locus, Aχ, is the complement of the semistable locus inX. Let X be proper or affine. There exists a fan in GR with support theset of characters in G with Xss 6= ∅. For each χ ∈ G, we have a cone

Cχ = {µ ∈ GR : Aµ ⊂ Aχ}.

These are the cones of the fan. The characters on the relative interiorsof the cones have equal unstable loci.

The maximal cones in the GIT fan are called the chambers. Thecodimension one cones are called walls.

If G is Abelian, the GIT fan is the GKZ fan.


Background on GIT

Hirzebruch surfaces

We can realize Fn as a GIT quotient of A4 by the subgroup

{(r, r−ns, r, s) : r, s ∈ Gm} ⊂ G4m.

Write k[x, y, u, v] for the ring of regular functions on A4.


Background on GIT

Hirzebruch surfaces

The GIT fan for this quotient is

x, u

vy

Fn

P(1, 1, n)

We have labeled rays by the variables with that associated characterand labeled the chambers according to their toric stacks.


Background on GIT

Stratifying the unstable locus

Let λ : Gm → G be a one parameter subgroup of G. Let L(λ) be thecentralizer of λ in G. Let Z denote the λ-fixed locus. For simplicity,assume Z is connected. The G-invariant subvariety, Z, inherits anL(λ)/λ(Gm)-action and an induced linearization (pulling back L). LetZλ denote the semi-stable locus and let S±λ be

S+λ := {x ∈ X | lim

t→∞λ(t) · x ∈ Zλ}

S−λ := {x ∈ X | limt→0

λ(t) · x ∈ Zλ}.

Denote by G · S±λ the orbit of S±λ under the G-action.


Background on GIT

Stratifying the unstable locus

Let X be a smooth projective variety equipped with the action ofreductive algebraic group, G. Choose an ample line bundle, L, with anequivariant structure.

Theorem (Kempf, Hesselink, Kirwan, Ness)There exist finitely many one-parameter subgroups,λi : Gm → G,with

Xus(L) = G · S+λ1∪ · · · ∪ G · S+

λp.


General results

Outline


2 Background on GIT

3 General results




General results

Setup

Suppose that we have a one-parameter family of linearizations, Lt,such that

Xss(L0) = G · S+λ1∪ · · · ∪ G · S+

λp∪ Xss(Lt) for t > 0

Xss(L0) = G · S−λ1∪ · · · ∪ G · S−λp

∪ Xss(Lt) for t < 0.

For example, X is proper or X is affine space and G is Abelian.Denote the quotient by t > 0 as X//+ and denote the quotient byt < 0 as X//−. If p = 1, let Y be the GIT quotient associated to Zλ1 .Choose a fixed point x ∈ Zλi . Let µi be the sum of the weights ofGm-action on the normal bundles to G · S+

λiand G · S−λi

restricted to x.


General results

Main theorem

Theorem (Ballard-F-Katzarkov, Halpern-Leinster)Fix d1, . . . , dp ∈ Z.

If µi > 0 for all 1 ≤ i ≤ p, then there exists a left-admissiblefully-faithful functor,

Φd1,...,dp : Db(coh X//−)→ Db(coh X//+).

If p = 1, then there also exists fully-faithful functors,

Υ−j : Db(coh Y)→ Db(coh X//+),

and a semi-orthogonal decomposition,

Db(coh X//+) = 〈Υ−−d Db(coh Y), . . . ,

Υ−µ−d−1 Db(coh Y),Φd Db(coh X//−)〉.


General results

Main theorem


If µi = 0 for all 1 ≤ i ≤ p, then there exist an equivalence,

Φd1,...,dp : Db(coh X//−)→ Db(coh X//+).


General results

Main theorem


If µ < 0 for all 1 ≤ i ≤ p, then there exists a left-admissiblefully-faithful functor,

Ψd1,...,dp : Db(coh X//+)→ Db(coh X//−)

If p = 1, then there also exists fully-faithful functors,

Υ+j : Db(coh Y)→ Db(coh X//+),


Db(coh X//−) = 〈Υ+−d Db(coh Y), . . . ,

Υ+µ−d+1 Db(coh Y),Ψd Db(coh X//+)〉.


General results

Simplifications

If the stabilizer of x is Gm, then µi > 0 if and only if thecanonical linearization lies on the positive plane for theseparating hyperplane corresponding to the wall (normalized sothat the + chamber is positive).

In the toric case i.e. if X is the Cox ring of a toric variety, X, andG = Pic(X), then a separating hyperplane is given by pairingwith a one parameter subgroup, λ, explicitly, 〈λ,−〉 is anelement of Pic(X)∗R and we choose λ to be primitive inHom(Pic(X),Z). In this case, p = 1 corresponding to this1-parameter subgroup and we get the strongest possible result.Furthermore, µ = 〈λ,−KX〉.


General results

Hirzebruch surfaces

The GIT fan for this quotient is

x, u

vy

Fn

P(1, 1, n)

We have labeled rays by the variables with that associated characterand labeled the chambers according to their toric stacks.


General results

Hirzebruch surfaces

• If n < 2, there is a semi-orthogonal decomposition

Db(coh Fn) = 〈E1, . . . ,E2−n,Db(coh P(1, 1, n))〉

with Ei exceptional objects.• If n = 2, we have an equivalence

Db(coh Fn) = Db(coh P(1, 1, 2)).

• If n > 2, there is a semi-orthogonal decomposition

Db(coh P(1, 1, n)) = 〈E1, . . . ,En−2,Db(coh Fn)〉

with Ei exceptional objects.


General results

Projective space/ bundles

Let B be a quasi-projective algebraic variety, and E be a vector bundleover B. Consider the Gm-action on the total space of E given bydilating the fibers. There are two linearizations of trivial bundlecorresponding to the identity character and inverse. Thecorresponding GIT-quotients are P(E) and the empty set. In this caseY ∼= B and we get:

Db(coh P(E)) = 〈π∗Db(coh B), . . . , π∗Db(coh B)(Oπ(n− 1))〉.

This explains why the Hirzebruch surface and weighted projectivestacks of the previous example decompose further.


General results

A theorem of Kawamata

Theorem

Let X be a smooth projective toric DM stack. Then Db(coh X) admitsa full exceptional collection.

Idea of Proof:


General results


Theorem


Idea of Proof: Let Σ0 denote the chamber corresponding to the nefcone of X. Choose a sufficiently straight line path in the secondary fanstarting at the anti-canonical divisor, passing through Σ0 and endingin the complement of the pseudo-effective cone which doesn’t passingthrough any cones of codimension 2.


General results


Theorem


Idea of Proof:

x, u

vy

Fn

P(1, 1, n)


General results


Theorem


Idea of Proof:The path, l, passes through a finite number of walls, τ1, . . . , τs on itsway through the “chambers”, Σ0, . . . ,Σs−1 and finally to the comple-ment. The value of µ is a pairing of the wall with the anti-canonicalclass, which is always positive because the anti-canonical class is al-ways behind you along this path. where Σs denotes the complement ofthe secondary fan as opposed to a chamber, thus the quotation marks.Let Xi be the GIT quotient corresponding to the chamber Σi withX := X0 and Xs = ∅ so that Db(coh Xs) := 0. Let Yi denote theGIT quotients coming from the walls as appearing the theorem.


General results


Theorem


Idea of Proof:We obtain a SOD decomposition:

Db(coh Xi) = 〈Db(coh Xi−1),Db(coh Yi), . . . ,Db(coh Yi)(µi − 1)〉.

Hence, combining these SODs, we obtain:

Db(coh X) = 〈Db(coh Y1), . . .Db(coh Y1), . . . ,Db(coh Ys), . . . ,Db(coh Ys)〉.


General results


Theorem


Idea of Proof:


Now, dim(Yi) < dim(X) for all i. Arguing by induction on the dimen-sion, we may assume that Db(coh Yi) admits a full exceptional collec-tion for all i. The base case is a point. Following the inductive processand the functors involved in the theorem, everything can be done withsheaves.


General results

Moduli Spaces

TheoremVarious compactifications of the moduli space of Pm with n distinctmarked points admit full exceptional collections.

Idea of Proof:


General results

Moduli Spaces(0, 0, 1)

(0, 1, 0)(1, 0, 0) (1/2, 1/2, 0)

(0, 1/2, 1/2)(1/2, 0, 1/2)

Σx Σy

Σz

Σ

GIT fan, n = 3, m = 1, intersection with the hyperplane, w1 + w2 + w3 = 1 inside Pic(P1 × P1 × P1) by PSL(2).


General results

Moduli Spaces


Idea of Proof: Let Σ0 denote the chamber corresponding to a givencompactification of the moduli space of n-points on Pm. Choose asufficiently straight line path in the secondary fan starting at the anti-canonical divisor, passing through Σ0 and ending in a chamber forwhich the GIT quotient is empty which doesn’t passing through anycones of codimension 2.


General results

Moduli Spaces


Idea of Proof:The path, l, passes through a finite number of walls, τ1, . . . , τs onits way through the chambers, Σ0, . . . ,Σs. The value of µ is alwayspositive because the anti-canonical class is always behind you alongthis path (by one of our simplifications). Let Xi be the GIT quotientcorresponding to the chamber Σi with X := X0 and Xs = ∅ so thatDb(coh Xs) := 0. Let Yi denote the GIT quotients coming from thewalls as appearing the theorem.


General results

Moduli Spaces


Idea of Proof:We obtain a SOD decomposition:

Db(coh Xi) = 〈Db(coh Xi−1),Db(coh Yi), . . . ,Db(coh Yi)(µi − 1)〉.

Hence, combining these SODs, we obtain:



General results

Moduli Spaces


Idea of Proof:


All the Yi are points, so we get an exceptional collection.


Landau-Ginzburg models and factorizations

Outline


2 Background on GIT

3 General results





Gauged LG-models

X is a smooth quasi-projective variety over an algebraicallyclosed field, k of characteristic zero,

G is a linearly reductive algebraic group acting on X,

L is a G-equivariant ample line bundle on X,

w is a G-invariant section of L.

DefinitionA gauged Landau-Ginzburg model (gauged LG-model) is thequadruple, (X,G,L,w), with X, G, L, and w as above.



Factorizations“Coherent sheaves” on a gauged LG-model, (X,G,L,w) are called fac-torizations.



Factorizations“Coherent sheaves” on a gauged LG-model, (X,G,L,w) are called fac-torizations.

DefinitionA factorization of a gauged LG-model, (X,G,L,w), consists of apair of coherent G-equivariant sheaves, E−1 and E0, and a pair ofG-equivariant OX-module homomorphisms,

φ−1E : E0 ⊗ L−1 → E−1

φ0E : E−1 → E0

such that the compositions, φ0E ◦ φ

−1E : E0 ⊗ L−1 → E0 and

φ−1E ⊗L◦φ0

E : E−1 → E−1⊗L, are isomorphic to multiplication by w.



Factorizations

DefinitionA factorization of a gauged LG-model, (X,G,L,w), consists of apair of coherent G-equivariant sheaves, E−1 and E0, and a pair ofG-equivariant OX-module homomorphisms,

φ−1E : E0 ⊗ L−1 → E−1

φ0E : E−1 → E0

such that the compositions, φ0E ◦ φ

−1E : E0 ⊗ L−1 → E0 and

φ−1E ⊗L◦φ0

E : E−1 → E−1⊗L, are isomorphic to multiplication by w.

The category of factorizations is an Abelian category akin to the cat-egory of complexes of coherent sheaves. An appropriate (dg) local-ization of this category by “acyclic factorizations” yields the derivedcategory of matrix factorizations, MF([X/G],w).



Plus a potential and auxiliary group action

We can also add in a potential, i.e. a G-invariant regular function,w ∈ Γ(X), and an auxiliary group action, H for which w isH-invariant. Assume, in addition to the previous conditions on thelinearization, that Xss(L0) admits an G× H-invariant affine cover. Welet X//± be the quotient by G× H, using on the semi-stable locus ofG. Similarly, Y is now the further quotient by H. w induces sectionsof line bundles on X//± and Y which we will denote w± and wY .



Main theorem

Theorem (Ballard-F-Katzarkov)Fix d1, . . . , dp ∈ Z.

If µi > 0 for 1 ≤ i ≤ p, then there exists a left-admissiblefully-faithful functor,

Φd1,...,dp : MF(X//−,w−)→ MF(X//+,w+)

If p = 1, then there exists fully-faithful functors,

Υ−j : MF(Y,wY)→ MF(X//+,w+),


MF(X//+,w+) = 〈Υ−−µ−d+1 MF(Y,wY), . . . ,

Υ−−d MF(Y,wY),Φd D(coh X//−,w−)〉.



Main theorem

Theorem (Ballard-F-Katzarkov)Fix d1, . . . , dp ∈ Z.

If µi = 0 for 1 ≤ i ≤ p, then there exist an equivalence,

Φd1,...,dp : MF(X//−,w−)→ MF(X//+,w+).



Main theorem

Theorem (Ballard-F-Katzarkov)Fix d ∈ Z.

If µi < 0 for 1 ≤ i ≤ p, then there exist fully-faithful functors,

Ψd1,...,dp : MF(X//+,w+)→ MF(X//−,w−)

If p = 1, then there exists fully-faithful functors,

Υ+j : MF(Y,wY)→ MF(X//+,w+),


MF(X//−,w−) = 〈Υ+−d MF(Y,wY), . . . ,

Υ+µ−d+1 MF(Y,wY),Ψd MF(X//+,w+)〉.


RG-flow and a theorem of Orlov

Outline


2 Background on GIT

3 General results





RG-flow

Theorem (Isik, Shipman)Let X be a variety and let σ : OX → E be a section of a vector bundle,E. Let Z denote the zero locus of σ and assume that all components ofZ have codimension equal to the rank of E. There is an equivalence

Db(coh Z) ∼= MF(tot E∨,w,Gm)

where w is the regular function induced by σ and the Gm is thedilation action on the fibers of tot E∨.

MF(X//+,w+) MF(X//−,w−)

Db(coh Z)

VGIT

RG-flowOrlov



Orlov’s Theorem

The following concept comes from work of Herbst, Hori, and Page,and was described mathematically in the Calabi-Yau case by Segaland Shipman. Consider a hyper surface Z, in P(V) defined byf ∈ H0(O(d)). By Isik/Shipman’s theorem we have:

Db(coh Z) ∼= MF(totO(−d), f ,Gm).

Now we want to do VGIT to reproduce a theorem of Orlov. Considerthe Gm-action on C× V with weights, −d and 1. There are two GITquotients: totO(−d) and V . Applying our main result we obtain:



Orlov’s Theorem (hypersurface/commutative case)

Let R = Sym(V).

Theorem (Orlov, hypersurface/commutative case)1 If n + 1− d > 0, there is a semi-orthogonal decomposition,

Db(coh Z) = 〈OZ(d − n), ...,OZ,MF(R, f ,Z)〉.

2 If n + 1− d = 0, there is an equivalence of triangulatedcategories,

Db(coh Z) = 〈MF(R, f ,Z)〉.3 If n + 1− d < 0, there is a semi-orthogonal decomposition,

MF(R, f ,Z) ∼=⟨k, . . . , k(n + 2− d),Db(coh Z)

⟩.



A generalization of Herbst and Walcher










Let X be a smooth toric variety, D1, ...,Ds be divisors, and E :=⊕si=1O(Di). Suppose fi are section of O(Di) forming a complete in-

tersection, Z, and let w be the corresponding map on E∨. We have,

Db(coh Z) ∼= MF(tot E∨,w,Gm).



A generalization of Herbst and WalcherLet X be a smooth toric variety, D1, ...,Ds be divisors, and E :=⊕s

i=1O(Di). Suppose fi are section of O(Di) forming a complete in-tersection, Z, and let w be the corresponding map on E∨. We have,

Db(coh Z) ∼= MF(tot E∨,w,Gm).

Now suppose the classes of the Di are all nef and lie on a wall sepa-rating the nef cone, Σ, from some other chamber of the GKZ fan, Σ′.Let X′ be the GIT quotient corresponding to the chamber Σ′. Lookinginstead at the secondary fan of E∨, if turns out that the pullback of Σand Σ′ are also chambers in this secondary fan, and the GIT quotientsare both E∨, once as a bundle over X and the other as a bundle overX′.



A generalization of Herbst and WalcherNow suppose the classes of the Di are all nef and lie on a wall sepa-rating the nef cone, Σ, from some other chamber of the GKZ fan, Σ′.Let X′ be the GIT quotient corresponding to the chamber Σ′. Lookinginstead at the secondary fan of E∨, if turns out that the pullback of Σand Σ′ are also chambers in this secondary fan, and the GIT quotientsare both E∨, once as a bundle over X and the other as a bundle over X′.We get two corresponding complete intersections and equivalences,

Db(coh Z) ∼= MF(tot E∨X ,w,Gm),

andDb(coh Z′) ∼= MF(tot E∨X′ ,w,Gm).

Let Y be the GIT quotient obtained by taking a generic character inthe wall, and thinking of the wall itself as a secondary fan of a toricvariety.



A generalization of Herbst and WalcherWe get two corresponding complete intersections and equivalences,

Db(coh Z) ∼= MF(tot E∨X ,w,Gm),

andDb(coh Z′) ∼= MF(tot E∨X′ ,w,Gm).

Let Y be the GIT quotient obtained by taking a generic character inthe wall, and thinking of the wall itself as a secondary fan of a toricvariety.




Theorem (Herbst-Walcher, Ballard-F-Katzarkov)1 If µ > 0, we have a SOD of Db(coh Z),

〈MF(Y,wY)(−µ− d + 1), . . . ,MF(Y,wY)(−d),Db(coh Z′)〉,

2 If µ = 0,Db(coh Z) = Db(coh Z′).

3 If µ < 0, we have a SOD of Db(coh Z′)

〈MF(Y,wY)(−d), . . . ,MF(Y,wY)(µ− d + 1),Db(coh Z)〉.



Homological Projective Duality

D(cohY,w) D(cohY ′,w)

Db(cohX ) Db(cohX ′)

VGIT

RG-flow RG-flow

HPD

Orlov



HPD exampleLet Li be a collection of ample line bundles on B for 1 ≤ i ≤ r,and consider the vector bundle, E =

⊕ri=1 Li. Let X be the projec-

tive bundle, π : P(E) → B and V := H0(E). The relative bundle,Oπ(1), provides an embedding, j : X → P(V∗). The universal hy-perplane section, X ⊂ X × P(V), is the zero section of a section,s ∈ Oπ(1) �O(1) (which we will explicitly describe later on). Let Ybe the total space ofOπ(−1)�O(−1) quotiented by fiberwise dilationby Gm. Employing Isik/Shipman’s Theorem we obtain an equivalenceDb(cohX ) ∼= D(cohY,w). We now describe a variation of lineariza-tion on E ×P(V)×A1 which yields, Y , on the one hand, and E ×P(V)on the other.



HPD exampleConsider the Gm-action on E∗×P(V)×A1, given by dilating the fibersof E∗, inverted dilation on A1, and acting trivially on P(V). Add anauxiliary Gm-action given by the usual action of Gm on A1 and actingtrivially on the other components. Consider the trivial line bundle, sothat the GIT quotients are determined by a character of Gm i.e. aninteger.



HPD exampleConsider the Gm-action on E∗×P(V)×A1, given by dilating the fibersof E∗, inverted dilation on A1, and acting trivially on P(V). Add anauxiliary Gm-action given by the usual action of Gm on A1 and actingtrivially on the other components. Consider the trivial line bundle, sothat the GIT quotients are determined by a character of Gm i.e. aninteger. Let Y ′ denote the global quotient stack of E∗ × P(V) by thefiberwise dilation by Gm. From this description, it follows that,

(E∗ × P(V)× A1)//+ = Y ,

(E∗ × P(V)× A1)//− = Y ′,Y = Z × P(V)× 0 ∼= B× P(V),

where in the GIT quotient we mod out by the auxiliary action aswell.



HPD exampleLet p : E∗ × P(V) × A1 → P(V) be the projection to P(V) and q :E∗×P(V)×A1 → E∗ be the projection to E . We can define a potential,w ∈ H0(p∗OP(V)(1)) by the pairing of V with V∗. Let u be a globalcoordinate on A1 so that, restricting to this component of E∗×P(V)×A1, g · x = g−1x for g ∈ Gm. Identifying H0(OP(V)(1)) with V∗ andthinking of V as global functions on E∗, let ei,j, . . . , ei,j be a basis forH0 (Lj) so that ranging over all i, j we obtain a basis of V , then thepotential is explicitly,

w := u∑

i,j

p∗e∗i,jq∗ei,j.



HPD exampleLet π1 : Y ′ → E∗ and π2 : Y ′ → P(V∗) be the two projections. Onechecks that,

w+ = 〈s,−〉,w− =

∑i,j π∗1e∗i,jπ∗2ei,j,

wY = 0.



HPD exampleLet π1 : Y ′ → E∗ and π2 : Y ′ → P(V∗) be the two projections. Onechecks that,

w+ = 〈s,−〉,w− =

∑i,j π∗1e∗i,jπ∗2ei,j,

wY = 0.

With L = Oπ(1) �OP(V)(1), by our main theorem, we obtain a semi-orthogonal decomposition,

Db(cohY,w+) = 〈Db(cohY ′,w−),Db(coh B×P(V)), . . . ,Db(coh B×P(V))(L⊗r−1)〉.



HPD exampleWith L = Oπ(1) �OP(V)(1), by our main theorem, we obtain a semi-orthogonal decomposition,

Db(cohY,w+) = 〈Db(cohY ′,w−),Db(coh B×P(V)), . . . ,Db(coh B×P(V))(L⊗r−1)〉.

Now, examining the explicit description of the potential, w−, we seethat it is the same as pairing with the section,

t =r⊕

i=1

∑j

e∗i,jei,j ∈r⊕

i=1

Li �OP(V)(1).

The zero set of this section is the complete intersection of zero loci of∑j e∗i,jei,j,

X ′ :=r⋂

i=1

Z(∑

j

e∗i,jei,j).



HPD exampleNow, examining the explicit description of the potential, w−, we seethat it is the same as pairing with the section,

t =r⊕

i=1

∑j

e∗i,jei,j ∈r⊕

i=1

Li �OP(V)(1).

The zero set of this section is the complete intersection of zero loci of∑j e∗i,jei,j,

X ′ :=r⋂

i=1

Z(∑

j

e∗i,jei,j).

Employing RG-flow here yields, D(cohY ′,w−) ∼= Db(cohX ′). Piec-ing this all together gives Db(cohX ) =

〈Db(cohX ′),Db(coh B×P(V)), . . . ,Db(coh B×P(V))((r−2)Oπ(1)�OP(V)(1)))〉.

The variety, X ′, is a homological projective dual of X.David Favero VGIT and Derived Categories


HPD exampleEmploying RG-flow here yields, D(cohY ′,w−) ∼= Db(cohX ′). Piec-ing this all together gives Db(cohX ) =

〈Db(cohX ′),Db(coh B×P(V)), . . . ,Db(coh B×P(V))((r−2)Oπ(1)�OP(V)(1)))〉.

The variety, X ′, is a homological projective dual of X. Consider thecomposition, f : X ′ ↪→ B × P(V) → P(V). A point p ∈ P(V),corresponds to a collection of sections, si ∈ H0(B,Li) for i = 1, . . . , rup to rescaling. The fiber of f over p, is the intersection,

⋂ri=1 Z(si) ⊆

B. When r = s + 1, the image of f is precisely the so called resultantvariety, consisting of points corresponding to degenerate collectionsof sections si ∈ H0(B,Li) i.e. collections of sections with nonemptyintersection. The resultant variety is the “classical projective dual”i.e. the set of singular hyperplane sections of X in P(V)


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Variation of Geometric Invariant Theory and Derived Categoriesfavero/slides/VGIT.pdf · 2020-05-08 · Variation of Geometric Invariant Theory and Derived Categories David Favero