Homological Mirror Symmetry and VGIT David Favero University of - - PowerPoint PPT Presentation

Homological Mirror Symmetry and VGIT

David Favero

University of Vienna

January 24, 2013

David Favero VGIT and Derived Categories

Attributions

Based on joint work with M. Ballard (U. Wisconsin) and Ludmil Katzarkov (U. Miami and U. Vienna). Slides available at http://www.mat.univie.ac.at/˜favero/slides/HMS.pdf

David Favero VGIT and Derived Categories

Calabi-Yau manifolds

Definition A Calabi-Yau (CY) manifold is a compact, simply connected K¨ ahler manifold (X, J) such that KX = det T∗X is trivial. Such a manifold possesses a closed (1, 1)-form B + iω, a complexified K¨ ahler form. H1(X, TX) parametrizes deformations of the complex structure H1(X, Ω1

X) parametrizes deformations of the complexified

symplectic structure

David Favero VGIT and Derived Categories

Hodge Diamonds for CY 3-folds

The Hodge diamond of a Calabi-Yau Threefold really only depends

• n

H1(X, TX) = H2,1(X) parametrizing deformations of the complex structure H1(X, Ω1

X) = H1,1(X) parametrizing deformations of the

complexified symplectic structure

David Favero VGIT and Derived Categories

Distribution of Calabi-Yau Threefolds

Plotted vertically is h1,1 + h2,1 Plotted horizontally is the Euler characteristic, χ(X) = 2(h1,1 − h2,1).

David Favero VGIT and Derived Categories

The A-Model and the B-Model

From physics: a Calabi-Yau variety with a complexified K¨ ahler class is supposed to give an N = 2 superconformal field theory. Given a N = 2 superconformal field theory, Witten proposed two “topologically twisted” field theories, the A-model and the B-model. These give rise to topological quantum field theories which are functors from the bordism category to a category of boundary conditions by the Calabi Yau.

David Favero VGIT and Derived Categories

The A-Model and the B-Model

For the A-model this category of boundary conditions is the Fukaya category of X. Objects are Langragian submanifolds and morphisms, are roughly, given by Floer cohomology complexes. This only depends on the symplectic structure of X. For the B-model this category of boundary conditions is the bounded derived category of coherent sheaves on X. Objects are complexes of coherent sheaves on X, morphisms are maps between complexes localized along maps which are isomorphisms on cohomology.

David Favero VGIT and Derived Categories

Homological Mirror Symmetry

Conjecture[Kontsevitch] For any Calabi-Yau manifold X, there exists a mirror X and equivalences of categories: Fuk(X) Fuk( X) Db(coh X) Db(coh X)

David Favero VGIT and Derived Categories

Gauged LG-models

Homological Mirror Symmetry can be extended beyond the world of Calabi-Yau manifolds if we allow our mirrors to be more exotic theories. Definition A Landau-Ginzburg model, (X, f) is a K¨ ahler manifold, X, together with a holomorphic function, w : X → A1.

David Favero VGIT and Derived Categories

A-model for an LG-model

For a Landau-Ginzburg model w : X → A1. with Morse singularities we can associate the Fukaya-Seidel category. Fix a smooth fiber of w and an ordering on the singular fibers. Objects are Lagrangrian thimbles. Let Ai and Aj be thimbles which degenerate to the ith and jth fiber respectively. Morphisms from Ai to Aj are given roughly by Floer cohomology if i ≤ j and are 0 otherwise.

David Favero VGIT and Derived Categories

B-model for an LG-model

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

E

: E0 → E−1 φ0

E : E−1 → E0

such that the compositions, φ0

E ◦ φ−1 E

: E0 → E0 and φ−1

E

• φ0

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

David Favero VGIT and Derived Categories

Homological Mirror Symmetry and LG-models

Conjecture[Generalized Homological Mirror Symmetry] For any Landau-Ginzburg-model (X, w) there exists a mirror (X, w) and equivalences of categories: Fuk(X, w) Fuk (X, w) Db(coh(X, w)) Db(coh (X, w))

David Favero VGIT and Derived Categories

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

Homological Mirror Symmetry and Birational Geometry (an example)

The mirror to P2 is the LG model (A2, x + y + 1

xy).

There are 3-singular fibers, each is an ordinary double point, and each gives a unique Lefschetz thimble up to isotopy. There is a semi-orthogonal decomposition Fuk(A2, x + y + 1 xy) = E1, E2, E3, where each of the Ei is equivalent to the simplest possible derived category, the category of graded vector spaces.

David Favero VGIT and Derived Categories

Homological Mirror Symmetry and Birational Geometry (an example)

The mirror to Blp P2 is the LG model (A2, x + y + xy + 1

xy).

There are 4-singular fibers, each is an ordinary double point, and each gives a unique Lefschetz thimble up to isotopy. There is a semi-orthogonal decomposition Fuk(A2, x + y + 1 xy) = E1, E2, E3, E4, where each of the Ei is equivalent to the simplest possible derived category, the category of graded vector spaces.

David Favero VGIT and Derived Categories

Mirror to Blp P2

✉ ✉ ✉ ✉ ❍ ❍ ✟ ✟ 3 ❍ ❍ ✟ ✟ 3

>

2

> > O T(-1) O(1) OEi

David Favero VGIT and Derived Categories

Homological Mirror Symmetry and Birational Geometry (an example)

Homological Mirror Symmetry was proven by Auroux, Katzarkov, and Orlov in this case. It predicts Db(coh P2) = Fuk(A2, x + y + 1 xy) = E1, E2, E3, where each of the Ei is equivalent to to the category of graded vector

• spaces. This is a theorem of Beilinson.

David Favero VGIT and Derived Categories

Homological Mirror Symmetry and Birational Geometry (an example)

On the other hand, Db(coh Blp P2) = Db(coh P2), E4 =E1, E2, E3, E4 = Fuk(A2, x + y + 1 x + 1 xy), where each of the Ei is equivalent to the category of graded vector spaces. Blowing-down a point corresponds to deforming the K¨ ahler form on Blp P2 or in the mirror it corresponds to deforming x + y + xy + 1

xy to

x + y + 1

xy.

In the mirror LG-model this corresponds to deforming the complex structure by changing the potential so that it has an additional singular fiber.

David Favero VGIT and Derived Categories

Blow-up - Blow-down

x, u v y Blp(P2) P2 Conclusion: Deforming the K¨ ahler form on the B-side yields semi-orthogonal decompositions!

David Favero VGIT and Derived Categories

Background on GIT

Given an action of G on X, one naturally wishes to form a “nice” quotient of X of G. Mumford shows us the non-canonical way. One chooses a G-equivariant line bundle and Mumford defines Xss(L) := {x ∈ X | ∃ f ∈ H0(X, Ln)G, f(x) = 0, and Xf affine} Xus(L) := X \ Xss(L). Xss(L) is the semi-stable locus and Xus(L) is the unstable locus.

David Favero VGIT and Derived Categories

The GIT quotient

Classically, the GIT quotient of X by G is the image of Xss(L) under the rational map X Proj

• n≥0

Γ(X, Ln)G. However, for us, the GIT quotient is the global quotient stack, [Xss(L)/G]. The classical GIT quotient is its coarse moduli space. We will let X/ /L denote the GIT quotient as a stack.

David Favero VGIT and Derived Categories

VGIT and Birational Geometry

Theorem (Hu, Keel) Let X → Y be a birational morphism between smooth projective varieties over C. There exists a smooth variety Z with a C∗-action, and an ample line bundle L with two linearizations L1 and L2 such that Z/ /L1 C∗ = X and Z/ /L2 C∗ = Y

David Favero VGIT and Derived Categories

Reminder on VGIT

By definition, GIT quotients depend on the choice of an ample G-equivariant line bundle. The parameter space for such quotients is then naturally the space of all G-equivariant ample line bundles, PicG(X)R. The unstable locus, Xχ, is the complement of the semistable locus in

• X. Let X be proper or affine. There exists a fan in PicG(X)R with

support the set of G-equivariant line bundles with Xss = ∅. For each L ∈ PicG(X)R, we have a cone CL = {L′ ∈ PicG(X)R : XL ⊂ XL′}. These are the cones of the fan.

David Favero VGIT and Derived Categories

Blow-up - Blow-down

We can realize Blp(P2) as a GIT quotient of A4 by the subgroup G2

m ∼

= {(r, r−1s, r, s) : r, s ∈ Gm} ⊂ G4

m.

Write k[x, y, u, v] for the ring of regular functions on A4. There are no nontrivial line bundles on A4. G-equivariant structures

• n the trivial bundle amount to characters of G2

m so our GIT fan lives

in a real plane.

David Favero VGIT and Derived Categories

Blow-up - Blow-down

The GIT fan for this quotient is x, u v y Blp(P2) P2

David Favero VGIT and Derived Categories

Wandering around on the K¨ ahler moduli space: the A-side

Mirror symmetry predicts that the K¨ ahler moduli space is exchanged with the moduli space of complex structures of the mirror. The GIT fan can be viewed as a piece of the K¨ ahler moduli space, or equivalently, a piece of the complex moduli space in the mirror. Hence, loops in this moduli space do not affect the A-model of the

• mirror. Instead, they give symplectic automorphisms of the mirror.

This piece of the complex moduli space of the mirror has nontrivial fundamental group after the discriminant locus of a type of universal hyperplane section is removed. Diemer, Katzarkov, and Kerr have shown that, mirror to our picture, variation of GIT for toric varieties provides symplectomorphisms and relations given by the combinatorial data. For example, they recover the lantern relation, the star relation, and Matsumota’s relations from the mapping class group of a Riemann surface.

David Favero VGIT and Derived Categories

Setup

The maximal cones in the GIT fan are called chambers. The codimension 1 cones are called walls. Let us say that X and Y are neighbors, if the lie in adjacent chambers separated by a wall. Work of Kirwan, Ness, Hesselink, and Kempf, tells us that the change in unstable locus between two neighbors is determined by a stratification given by a finite number of one parameter subgroups λ1, ..., λp of G. For each λi let µi be the difference of the weights of the determinants

• f the conormal bundles to the corresponding piece of stratification on

X and Y.

David Favero VGIT and Derived Categories

More Setup

For each one parameter subgroup λ : C∗ → G one can look at the fixed locus S0

λ.

There is a residual group action of C(λ)/λ the centralizer of λ modulo λ on S0

λ.

The restriction of our G-equivariant line bundle on X induces a C(λ)/λ-equivariant line bundle on S0

λ.

So we get a new GIT quotient of S0

λ by C(λ)/λ, call it Z! (actually, as

before it’s a stack)

David Favero VGIT and Derived Categories

Main theorem

Theorem (Ballard-F-Katzarkov, Halpern-Leinster) Let X and Y be neighbors. If µi > 0 for all 1 ≤ i ≤ p, then there exists a left-admissible fully-faithful functor, Φ : Db(coh X) → Db(coh Y). If p = 1, then there also exists fully-faithful functors, Υ−

j : Db(coh Z) → Db(coh X),

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

−d Db(coh Z), . . . , Υ− µ−d−1 Db(coh Z), Φd Db(coh Y).

David Favero VGIT and Derived Categories

Main theorem

Theorem (Ballard-F-Katzarkov, Halpern-Leinster) If µi = 0 for all 1 ≤ i ≤ p, then there exist an equivalence, Φ : Db(coh X) → Db(coh Y).

David Favero VGIT and Derived Categories

Main theorem

Theorem (Ballard-F-Katzarkov, Halpern-Leinster) If µ < 0 for all 1 ≤ i ≤ p, then there exists a left-admissible fully-faithful functor, Ψ : Db(coh Y) → Db(coh X) If p = 1, then there also exists fully-faithful functors, Υ+

j : Db(coh Z) → Db(coh Y),

and a semi-orthogonal decomposition, Db(coh Y) = Υ+

−d Db(coh Z), . . . , Υ+ µ−d+1 Db(coh Z), Ψd Db(coh X).

David Favero VGIT and Derived Categories

LG-models too!

The same theorem holds for LG-models (in fact we prove this in the generality of “gauged” LG-models, meaning factorizations which are equivariant with respect to the action of a group G.

David Favero VGIT and Derived Categories

Applications

Very pleasant inductive structure for the derived categories of projective toric DM stacks (recovers Kawamata) Similar inductive structure for derived categories of moduli of stable pointed rational curves (recovers Manin-Smirnov) and decompositions of rational Chow motives. Full generalization of the σ-model/Landau-Ginzburg correspondence (recovers Orlov in commutative case). New derived equivalences for birational varieties (recovers Herbst-Walcher, Kawamata, Van den Bergh, Orlov). New relationships between Chow Groups/ Griffiths groups of smooth varieties. Provides a framework for the mirror to symplectic automorphisms. Provides an explanation of Homological Mirror Symmetry for Toric Varieties.

David Favero VGIT and Derived Categories

That’s All Folks

The End.

David Favero VGIT and Derived Categories