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Variation of Geometric Invariant Theory and Derived 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 Ludmil Katzarkov (U. Miami and U. Vienna). Available at http://arxiv.org/abs/1203.6643.

David Favero VGIT and Derived Categories

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

David Favero VGIT and Derived Categories

Motivating Example

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

David Favero VGIT and Derived Categories

Motivating Example

Weighted Projective Stacks

Consider k[x0, x1, x2] with the Gm-action with weights (1, 1, n). We define P(1 : 1 : n) as the smooth global quotient Deligne-Mumford stack, [(Spec k[x0, x1, x2]\0)/Gm]. Characters of Gm, λ → λi, give line 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

David Favero VGIT and Derived Categories

Motivating Example

Hirzebruch Surfaces

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

• x2

x2 x0 x1 x1 x0 x3 x3

1

x2

0x1

x0x2

1

David Favero VGIT and Derived Categories

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 x3 x3

1

x2

0x1

x0x2

1

David Favero VGIT and Derived Categories

Motivating Example

Semi-orthogonal decompositions

Definition A semi-orthogonal decomposition of a triangulated category, T , is a sequence of full triangulated subcategories, A1, . . . , Am, in T such that Ai ⊂ A⊥

j for i < j and, for every object T ∈ T , there exists a

diagram: Tm−1 · · · T2 T1 T Am A2 A1

| | |

where all triangles are distinguished and Ak ∈ Ak. We denote a semi-orthogonal decomposition by A1, . . . , Am.

David Favero VGIT and Derived Categories

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.

David Favero VGIT and Derived Categories

Background on GIT

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

David Favero VGIT and Derived Categories

Background on GIT

Reminder on VGIT

X is a smooth quasi-projective variety over an algebraically closed 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) = 0, and Xf affine} For us, the GIT quotient corresponding to this data is the global quotient stack [Xss(L)/G]. We can vary the G-equivariant structure on L by choosing characters, χ, in the dual group, G := Hom(G, Gm). We denote the GIT quotient corresponding to this linearization by X/ /L(χ).

David Favero VGIT and Derived Categories

Background on GIT

Reminder on VGIT

The unstable locus, Aχ, is the complement of the semistable locus in

• X. Let X be proper or affine. There exists a fan in

GR with support the set of characters in G with Xss = ∅. 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 interiors

• f the cones have equal unstable loci.

The maximal cones in the GIT fan are called the chambers. The codimension one cones are called walls. If G is Abelian, the GIT fan is the GKZ fan.

David Favero VGIT and Derived Categories

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} ⊂ G4

m.

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

David Favero VGIT and Derived Categories

Background on GIT

Hirzebruch surfaces

The GIT fan for this quotient is x, u v y Fn P(1, 1, n) We have labeled rays by the variables with that associated character and labeled the chambers according to their toric stacks.

David Favero VGIT and Derived Categories

Background on GIT

Stratifying the unstable locus

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

λ be

S+

λ := {x ∈ X | lim t→∞ λ(t) · x ∈ Zλ}

S−

λ := {x ∈ X | lim t→0 λ(t) · x ∈ Zλ}.

Denote by G · S±

λ the orbit of S± λ under the G-action.

David Favero VGIT and Derived Categories

Background on GIT

Stratifying the unstable locus

Let X be a smooth projective variety equipped with the action of reductive algebraic group, G. Choose an ample line bundle, L, with an equivariant 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.

David Favero VGIT and Derived Categories

General results

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

David Favero VGIT and Derived Categories

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 by t < 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 of Gm-action on the normal bundles to G · S+

λi and G · S− λi restricted to x.

David Favero VGIT and Derived Categories

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-admissible fully-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/

/−).

David Favero VGIT and Derived Categories

General results

Main theorem

Theorem (Ballard-F-Katzarkov, Halpern-Leinster) Fix d1, . . . , dp ∈ Z. If µi = 0 for all 1 ≤ i ≤ p, then there exist an equivalence, Φd1,...,dp : Db(coh X/ /−) → Db(coh X/ /+).

David Favero VGIT and Derived Categories

General results

Main theorem

Theorem (Ballard-F-Katzarkov, Halpern-Leinster) Fix d1, . . . , dp ∈ Z. If µ < 0 for all 1 ≤ i ≤ p, then there exists a left-admissible fully-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/

/+), and a semi-orthogonal decomposition, Db(coh X/ /−) = Υ+

−d Db(coh Y), . . . ,

Υ+

µ−d+1 Db(coh Y), Ψd Db(coh X/

/+).

David Favero VGIT and Derived Categories

General results

Simplifications

If the stabilizer of x is Gm, then µi > 0 if and only if the canonical linearization lies on the positive plane for the separating hyperplane corresponding to the wall (normalized so that the + chamber is positive). In the toric case i.e. if X is the Cox ring of a toric variety, X, and G = Pic(X), then a separating hyperplane is given by pairing with a one parameter subgroup, λ, explicitly, λ, − is an element of Pic(X)∗

R and we choose λ to be primitive in

Hom(Pic(X), Z). In this case, p = 1 corresponding to this 1-parameter subgroup and we get the strongest possible result. Furthermore, µ = λ, −KX.

David Favero VGIT and Derived Categories

General results

Hirzebruch surfaces

The GIT fan for this quotient is x, u v y Fn P(1, 1, n) We have labeled rays by the variables with that associated character and labeled the chambers according to their toric stacks.

David Favero VGIT and Derived Categories

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.

David Favero VGIT and Derived Categories

General results

Projective space/ bundles

Let B be a quasi-projective algebraic variety, and E be a vector bundle

• ver B. Consider the Gm-action on the total space of E given by

dilating the fibers. There are two linearizations of trivial bundle corresponding to the identity character and inverse. The corresponding GIT-quotients are P(E) and the empty set. In this case Y ∼ = 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 projective stacks of the previous example decompose further.

David Favero VGIT and Derived Categories

General results

A theorem of Kawamata

Theorem Let X be a smooth projective toric DM stack. Then Db(coh X) admits a full exceptional collection. Idea of Proof:

David Favero VGIT and Derived Categories

General results

A theorem of Kawamata

Theorem Let X be a smooth projective toric DM stack. Then Db(coh X) admits a full exceptional collection. Idea of Proof: Let Σ0 denote the chamber corresponding to the nef cone of X. Choose a sufficiently straight line path in the secondary fan starting at the anti-canonical divisor, passing through Σ0 and ending in the complement of the pseudo-effective cone which doesn’t passing through any cones of codimension 2.

David Favero VGIT and Derived Categories

General results

A theorem of Kawamata

Theorem Let X be a smooth projective toric DM stack. Then Db(coh X) admits a full exceptional collection. Idea of Proof: x, u v y Fn P(1, 1, n)

David Favero VGIT and Derived Categories

General results

A theorem of Kawamata

Theorem Let X be a smooth projective toric DM stack. Then Db(coh X) admits a full exceptional collection. Idea of Proof: The path, l, passes through a finite number of walls, τ1, . . . , τs on its way 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-canonical

class, which is always positive because the anti-canonical class is al- ways behind you along this path. where Σs denotes the complement of the secondary fan as opposed to a chamber, thus the quotation marks. Let Xi be the GIT quotient corresponding to the chamber Σi with X := X0 and Xs = ∅ so that Db(coh Xs) := 0. Let Yi denote the GIT quotients coming from the walls as appearing the theorem.

David Favero VGIT and Derived Categories

General results

A theorem of Kawamata

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

David Favero VGIT and Derived Categories

General results

A theorem of Kawamata

Theorem Let X be a smooth projective toric DM stack. Then Db(coh X) admits a full exceptional collection. Idea of Proof:

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

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 process and the functors involved in the theorem, everything can be done with sheaves.

David Favero VGIT and Derived Categories

General results

Moduli Spaces

Theorem Various compactifications of the moduli space of Pm with n distinct marked points admit full exceptional collections. Idea of Proof:

David Favero VGIT and Derived Categories

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). David Favero VGIT and Derived Categories

General results

Moduli Spaces

Theorem Various compactifications of the moduli space of Pm with n distinct marked points admit full exceptional collections. Idea of Proof: Let Σ0 denote the chamber corresponding to a given compactification of the moduli space of n-points on Pm. Choose a sufficiently straight line path in the secondary fan starting at the anti- canonical divisor, passing through Σ0 and ending in a chamber for which the GIT quotient is empty which doesn’t passing through any cones of codimension 2.

David Favero VGIT and Derived Categories

General results

Moduli Spaces

Theorem Various compactifications of the moduli space of Pm with n distinct marked points admit full exceptional collections. Idea of Proof: The path, l, passes through a finite number of walls, τ1, . . . , τs on its way through the chambers, Σ0, . . . , Σs. The value of µ is always positive because the anti-canonical class is always behind you along this path (by one of our simplifications). Let Xi be the GIT quotient corresponding to the chamber Σi with X := X0 and Xs = ∅ so that Db(coh Xs) := 0. Let Yi denote the GIT quotients coming from the walls as appearing the theorem.

David Favero VGIT and Derived Categories

General results

Moduli Spaces

Theorem Various compactifications of the moduli space of Pm with n distinct marked points admit full exceptional collections. 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).

David Favero VGIT and Derived Categories

General results

Moduli Spaces

Theorem Various compactifications of the moduli space of Pm with n distinct marked points admit full exceptional collections. Idea of Proof:

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

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

David Favero VGIT and Derived Categories

Landau-Ginzburg models and factorizations

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

David Favero VGIT and Derived Categories

Landau-Ginzburg models and factorizations

Gauged LG-models

X is a smooth quasi-projective variety over an algebraically closed 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. Definition A gauged Landau-Ginzburg model (gauged LG-model) is the quadruple, (X, G, L, w), with X, G, L, and w as above.

David Favero VGIT and Derived Categories

Landau-Ginzburg models and factorizations

Factorizations

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

David Favero VGIT and Derived Categories

Landau-Ginzburg models and factorizations

Factorizations

“Coherent sheaves” on a gauged LG-model, (X, G, L, w) are called fac- torizations. Definition A factorization of a gauged LG-model, (X, G, L, w), consists of a pair of coherent G-equivariant sheaves, E−1 and E0, and a pair of G-equivariant OX-module homomorphisms, φ−1

E

: E0 ⊗ L−1 → E−1 φ0

E : E−1 → E0

such that the compositions, φ0

E ◦ φ−1 E

: E0 ⊗ L−1 → E0 and φ−1

E ⊗L◦φ0 E : E−1 → E−1 ⊗L, are isomorphic to multiplication by w.

David Favero VGIT and Derived Categories

Landau-Ginzburg models and factorizations

Factorizations

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

E

: E0 ⊗ L−1 → E−1 φ0

E : E−1 → E0

such that the compositions, φ0

E ◦ φ−1 E

: E0 ⊗ L−1 → E0 and φ−1

E ⊗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 derived category of matrix factorizations, MF([X/G], w).

David Favero VGIT and Derived Categories

Landau-Ginzburg models and factorizations

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 is H-invariant. Assume, in addition to the previous conditions on the linearization, that Xss(L0) admits an G × H-invariant affine cover. We let X/ /± be the quotient by G × H, using on the semi-stable locus of

• G. Similarly, Y is now the further quotient by H. w induces sections
• f line bundles on X/

/± and Y which we will denote w± and wY.

David Favero VGIT and Derived Categories

Landau-Ginzburg models and factorizations

Main theorem

Theorem (Ballard-F-Katzarkov) Fix d1, . . . , dp ∈ Z. If µi > 0 for 1 ≤ i ≤ p, then there exists a left-admissible fully-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+), and a semi-orthogonal decomposition, MF(X/ /+, w+) = Υ−

−µ−d+1 MF(Y, wY), . . . ,

Υ−

−d MF(Y, wY), Φd D(coh X/

/−, w−).

David Favero VGIT and Derived Categories

Landau-Ginzburg models and factorizations

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+).

David Favero VGIT and Derived Categories

Landau-Ginzburg models and factorizations

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+), and a semi-orthogonal decomposition, MF(X/ /−, w−) = Υ+

−d MF(Y, wY), . . . ,

Υ+

µ−d+1 MF(Y, wY), Ψd MF(X/

/+, w+).

David Favero VGIT and Derived Categories

RG-flow and a theorem of Orlov

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

David Favero VGIT and Derived Categories

RG-flow and a theorem of Orlov

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 of

Z 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 the dilation action on the fibers of tot E∨. MF(X/ /+, w+) MF(X/ /−, w−) Db(coh Z) VGIT RG-flow Orlov

David Favero VGIT and Derived Categories

RG-flow and a theorem of Orlov

Orlov’s Theorem

The following concept comes from work of Herbst, Hori, and Page, and was described mathematically in the Calabi-Yau case by Segal and Shipman. Consider a hyper surface Z, in P(V) defined by f ∈ H0(O(d)). By Isik/Shipman’s theorem we have: Db(coh Z) ∼ = MF(tot O(−d), f, Gm). Now we want to do VGIT to reproduce a theorem of Orlov. Consider the Gm-action on C × V with weights, −d and 1. There are two GIT quotients: tot O(−d) and V. Applying our main result we obtain:

David Favero VGIT and Derived Categories

RG-flow and a theorem of Orlov

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 triangulated categories, 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)
• .

David Favero VGIT and Derived Categories

RG-flow and a theorem of Orlov

A generalization of Herbst and Walcher

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 of

Z 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 the dilation action on the fibers of tot E∨.

David Favero VGIT and Derived Categories

RG-flow and a theorem of Orlov

A generalization of Herbst and Walcher

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 of

Z 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 the dilation action on the fibers of tot E∨. Let X be a smooth toric variety, D1, ..., Ds be divisors, and E := s

i=1 O(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).

David Favero VGIT and Derived Categories

RG-flow and a theorem of Orlov

A generalization of Herbst and Walcher

Let X be a smooth toric variety, D1, ..., Ds be divisors, and E := s

i=1 O(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′. Looking instead at the secondary fan of E∨, if turns out that the pullback of Σ and Σ′ are also chambers in this secondary fan, and the GIT quotients are both E∨, once as a bundle over X and the other as a bundle over X′.

David Favero VGIT and Derived Categories

RG-flow and a theorem of Orlov

A generalization of Herbst and Walcher

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′. Looking instead at the secondary fan of E∨, if turns out that the pullback of Σ and Σ′ are also chambers in this secondary fan, and the GIT quotients are 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),

and Db(coh Z′) ∼ = MF(tot E∨

X′, w, Gm).

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

David Favero VGIT and Derived Categories

RG-flow and a theorem of Orlov

A generalization of Herbst and Walcher

We get two corresponding complete intersections and equivalences, Db(coh Z) ∼ = MF(tot E∨

X, w, Gm),

and Db(coh Z′) ∼ = MF(tot E∨

X′, w, Gm).

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

David Favero VGIT and Derived Categories

RG-flow and a theorem of Orlov

A generalization of Herbst and Walcher

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).

David Favero VGIT and Derived Categories

RG-flow and a theorem of Orlov

Homological Projective Duality

D(coh Y, w) D(coh Y′, w) Db(coh X) Db(coh X ′) VGIT RG-flow RG-flow HPD Orlov

David Favero VGIT and Derived Categories

RG-flow and a theorem of Orlov

HPD example

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

i=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 Y be the total space of Oπ(−1)⊠O(−1) quotiented by fiberwise dilation by Gm. Employing Isik/Shipman’s Theorem we obtain an equivalence Db(coh X) ∼ = D(coh Y, 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)

• n the other.

David Favero VGIT and Derived Categories

RG-flow and a theorem of Orlov

HPD example

Consider the Gm-action on E∗×P(V)×A1, given by dilating the fibers

• f E∗, inverted dilation on A1, and acting trivially on P(V). Add an

auxiliary Gm-action given by the usual action of Gm on A1 and acting trivially on the other components. Consider the trivial line bundle, so that the GIT quotients are determined by a character of Gm i.e. an integer.

David Favero VGIT and Derived Categories

RG-flow and a theorem of Orlov

HPD example

Consider the Gm-action on E∗×P(V)×A1, given by dilating the fibers

• f E∗, inverted dilation on A1, and acting trivially on P(V). Add an

auxiliary Gm-action given by the usual action of Gm on A1 and acting trivially on the other components. Consider the trivial line bundle, so that the GIT quotients are determined by a character of Gm i.e. an integer. Let Y′ denote the global quotient stack of E∗ × P(V) by the fiberwise 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 as well.

David Favero VGIT and Derived Categories

RG-flow and a theorem of Orlov

HPD example

Let 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 global coordinate 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∗ and thinking of V as global functions on E∗, let ei,j, . . . , ei,j be a basis for H0 (Lj) so that ranging over all i, j we obtain a basis of V, then the potential is explicitly, w := u

• i,j

p∗e∗

i,jq∗ei,j.

David Favero VGIT and Derived Categories

RG-flow and a theorem of Orlov

HPD example

Let π1 : Y′ → E∗ and π2 : Y′ → P(V∗) be the two projections. One checks that, w+ = s, −, w− =

i,j π∗ 1e∗ i,jπ∗ 2ei,j,

wY = 0.

David Favero VGIT and Derived Categories

RG-flow and a theorem of Orlov

HPD example

Let π1 : Y′ → E∗ and π2 : Y′ → P(V∗) be the two projections. One checks 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-

• rthogonal decomposition,

Db(coh Y, w+) = Db(coh Y′, w−), Db(coh B×P(V)), . . . , Db(coh B×P(V))(

David Favero VGIT and Derived Categories

RG-flow and a theorem of Orlov

HPD example

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

• rthogonal decomposition,

Db(coh Y, w+) = Db(coh Y′, w−), Db(coh B×P(V)), . . . , Db(coh B×P(V))( Now, examining the explicit description of the potential, w−, we see that 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).

David Favero VGIT and Derived Categories

RG-flow and a theorem of Orlov

HPD example

Now, examining the explicit description of the potential, w−, we see that 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(coh Y′, w−) ∼ = Db(coh X ′). Piec- ing this all together gives Db(coh X) =

Db(coh X ′), 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

RG-flow and a theorem of Orlov

HPD example

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

Db(coh X ′), 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 the composition, 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, . . . , r up to rescaling. The fiber of f over p, is the intersection, r

i=1 Z(si) ⊆

• B. When r = s + 1, the image of f is precisely the so called resultant

variety, consisting of points corresponding to degenerate collections

• f sections si ∈ H0(B, Li) i.e. collections of sections with nonempty
• intersection. The resultant variety is the “classical projective dual”

i.e. the set of singular hyperplane sections of X in P(V)

David Favero VGIT and Derived Categories