Homological Mirror Symmetry and VGIT


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1 Homological Mirror Symmetry and VGIT University of Vienna January 24, 2013
2 Attributions Based on joint work with M. Ballard (U. Wisconsin) and Ludmil Katzarkov (U. Miami and U. Vienna). Slides available at favero/slides/hms.pdf
3 CalabiYau manifolds Definition A CalabiYau (CY) manifold is a compact, simply connected Kähler manifold (X, J) such that K X = det T X is trivial. Such a manifold possesses a closed (1, 1)form B + iω, a complexified Kähler form. H 1 (X, T X ) parametrizes deformations of the complex structure H 1 (X, Ω 1 X ) parametrizes deformations of the complexified symplectic structure
4 Hodge Diamonds for CY 3folds The Hodge diamond of a CalabiYau Threefold really only depends on H 1 (X, T X ) = H 2,1 (X) parametrizing deformations of the complex structure H 1 (X, Ω 1 X ) = H1,1 (X) parametrizing deformations of the complexified symplectic structure
5 Distribution of CalabiYau Threefolds Plotted vertically is h 1,1 + h 2,1 Plotted horizontally is the Euler characteristic, χ(x) = 2(h 1,1 h 2,1 ).
6 The AModel and the BModel From physics: a CalabiYau variety with a complexified Kähler 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 Amodel and the Bmodel. 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.
7 The AModel and the BModel For the Amodel 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 Bmodel 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.
8 Homological Mirror Symmetry Conjecture[Kontsevitch] For any CalabiYau manifold X, there exists a mirror X and equivalences of categories: Fuk(X) Fuk( X) D b (coh X) D b (coh X)
9 Gauged LGmodels Homological Mirror Symmetry can be extended beyond the world of CalabiYau manifolds if we allow our mirrors to be more exotic theories. Definition A LandauGinzburg model, (X, f ) is a Kähler manifold, X, together with a holomorphic function, w : X A 1.
10 Amodel for an LGmodel For a LandauGinzburg model w : X A 1. with Morse singularities we can associate the FukayaSeidel category. Fix a smooth fiber of w and an ordering on the singular fibers. Objects are Lagrangrian thimbles. Let A i and A j be thimbles which degenerate to the i th and j th fiber respectively. Morphisms from A i to A j are given roughly by Floer cohomology if i j and are 0 otherwise.
11 Bmodel for an LGmodel Coherent sheaves on an LGmodel, (X, w) are called factorizations. Definition A factorization of an LGmodel, (X, w), consists of a pair of coherent sheaves, E 1 and E 0, and a pair of O X module homomorphisms, φ 1 E : E 0 E 1 φ 0 E : E 1 E 0 such that the compositions, φ 0 E φ 1 E : E 0 E 0 and φ 1 E φ 0 E : E 1 E 1, are isomorphic to multiplication by w.
12 Homological Mirror Symmetry and LGmodels Conjecture[Generalized Homological Mirror Symmetry] For any LandauGinzburgmodel (X, w) there exists a mirror (X, w) and equivalences of categories: Fuk(X, w) Fuk (X, w) D b (coh(x, w)) D b (coh (X, w))
13 Semiorthogonal decompositions Definition A semiorthogonal decomposition of a triangulated category, T, is a sequence of full triangulated subcategories, A 1,..., A m, in T such that A i A j for i < j and, for every object T T, there exists a diagram: 0 T m 1 T 2 T 1 T A m A 2 A 1 where all triangles are distinguished and A k A k. We denote a semiorthogonal decomposition by A 1,..., A m.
14 Homological Mirror Symmetry and Birational Geometry (an example) The mirror to P 2 is the LG model (A 2, x + y + 1 xy ). There are 3singular fibers, each is an ordinary double point, and each gives a unique Lefschetz thimble up to isotopy. There is a semiorthogonal decomposition Fuk(A 2, x + y + 1 xy ) = E 1, E 2, E 3, where each of the E i is equivalent to the simplest possible derived category, the category of graded vector spaces.
15 Homological Mirror Symmetry and Birational Geometry (an example) The mirror to Bl p P 2 is the LG model (A 2, x + y + xy + 1 xy ). There are 4singular fibers, each is an ordinary double point, and each gives a unique Lefschetz thimble up to isotopy. There is a semiorthogonal decomposition Fuk(A 2, x + y + 1 xy ) = E 1, E 2, E 3, E 4, where each of the E i is equivalent to the simplest possible derived category, the category of graded vector spaces.
16 Mirror to Bl p P 2 > > 2 > 3 3 O T(1) O(1) O Ei
17 Homological Mirror Symmetry and Birational Geometry (an example) Homological Mirror Symmetry was proven by Auroux, Katzarkov, and Orlov in this case. It predicts D b (coh P 2 ) = Fuk(A 2, x + y + 1 xy ) = E 1, E 2, E 3, where each of the E i is equivalent to to the category of graded vector spaces. This is a theorem of Beilinson.
18 Homological Mirror Symmetry and Birational Geometry (an example) On the other hand, D b (coh Bl p P 2 ) = D b (coh P 2 ), E 4 = E 1, E 2, E 3, E 4 = Fuk(A 2, x + y + 1 x + 1 xy ), where each of the E i is equivalent to the category of graded vector spaces. Blowingdown a point corresponds to deforming the Kähler form on Bl p P 2 or in the mirror it corresponds to deforming x + y + xy + 1 xy to x + y + 1 xy. In the mirror LGmodel this corresponds to deforming the complex structure by changing the potential so that it has an additional singular fiber.
19 Blowup  Blowdown y v P 2 Bl p (P 2 ) x, u Conclusion: Deforming the Kähler form on the Bside yields semiorthogonal decompositions!
20 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 noncanonical way. One chooses a Gequivariant line bundle and Mumford defines X ss (L) := {x X f H 0 (X, L n ) G, f (x) 0, and X f affine} X us (L) := X \ X ss (L). X ss (L) is the semistable locus and X us (L) is the unstable locus.
21 The GIT quotient Classically, the GIT quotient of X by G is the image of X ss (L) under the rational map X Proj n 0 Γ(X, L n ) G. However, for us, the GIT quotient is the global quotient stack, [X ss (L)/G]. The classical GIT quotient is its coarse moduli space. We will let X /L denote the GIT quotient as a stack.
22 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 L 1 and L 2 such that Z / L1 C = X and Z / L2 C = Y
23 Reminder on VGIT By definition, GIT quotients depend on the choice of an ample Gequivariant line bundle. The parameter space for such quotients is then naturally the space of all Gequivariant ample line bundles, Pic G (X) R. The unstable locus, X χ, is the complement of the semistable locus in X. Let X be proper or affine. There exists a fan in Pic G (X) R with support the set of Gequivariant line bundles with X ss. For each L Pic G (X) R, we have a cone These are the cones of the fan. C L = {L Pic G (X) R : X L X L }.
24 Blowup  Blowdown We can realize Bl p (P 2 ) as a GIT quotient of A 4 by the subgroup G 2 m = {(r, r 1 s, r, s) : r, s G m } G 4 m. Write k[x, y, u, v] for the ring of regular functions on A 4. There are no nontrivial line bundles on A 4. Gequivariant structures on the trivial bundle amount to characters of G 2 m so our GIT fan lives in a real plane.
25 Blowup  Blowdown The GIT fan for this quotient is y v P 2 Bl p (P 2 ) x, u
26 Wandering around on the Kähler moduli space: the Aside Mirror symmetry predicts that the Kähler 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 Kähler moduli space, or equivalently, a piece of the complex moduli space in the mirror. Hence, loops in this moduli space do not affect the Amodel 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.
27 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 λ 1,..., λ p of G. For each λ i let µ i be the difference of the weights of the determinants of the conormal bundles to the corresponding piece of stratification on X and Y.
28 More Setup For each one parameter subgroup λ : C G one can look at the fixed locus S 0 λ. There is a residual group action of C(λ)/λ the centralizer of λ modulo λ on S 0 λ. The restriction of our Gequivariant line bundle on X induces a C(λ)/λequivariant line bundle on S 0 λ. So we get a new GIT quotient of Sλ 0 by C(λ)/λ, call it Z! (actually, as before it s a stack)
29 Main theorem Theorem (BallardFKatzarkov, HalpernLeinster) Let X and Y be neighbors. If µ i > 0 for all 1 i p, then there exists a leftadmissible fullyfaithful functor, Φ : D b (coh X) D b (coh Y). If p = 1, then there also exists fullyfaithful functors, Υ j : D b (coh Z) D b (coh X), and a semiorthogonal decomposition, D b (coh X) = Υ d Db (coh Z),..., Υ µ d 1 Db (coh Z), Φ d D b (coh Y).
30 Main theorem Theorem (BallardFKatzarkov, HalpernLeinster) If µ i = 0 for all 1 i p, then there exist an equivalence, Φ : D b (coh X) D b (coh Y).
31 Main theorem Theorem (BallardFKatzarkov, HalpernLeinster) If µ < 0 for all 1 i p, then there exists a leftadmissible fullyfaithful functor, Ψ : D b (coh Y) D b (coh X) If p = 1, then there also exists fullyfaithful functors, Υ + j : D b (coh Z) D b (coh Y), and a semiorthogonal decomposition, D b (coh Y) = Υ + d Db (coh Z),..., Υ + µ d+1 Db (coh Z), Ψ d D b (coh X).
32 LGmodels too! The same theorem holds for LGmodels (in fact we prove this in the generality of gauged LGmodels, meaning factorizations which are equivariant with respect to the action of a group G.
33 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 ManinSmirnov) and decompositions of rational Chow motives. Full generalization of the σmodel/landauginzburg correspondence (recovers Orlov in commutative case). New derived equivalences for birational varieties (recovers HerbstWalcher, 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.
34 That s All Folks The End.
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