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 K X = det T ∗ X is trivial. Such a manifold possesses a closed ( 1 , 1 ) -form B + i ω , a complexified K¨ ahler 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 David Favero VGIT and Derived Categories
Hodge Diamonds for CY 3-folds The Hodge diamond of a Calabi-Yau 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 ) = H 1 , 1 ( X ) parametrizing deformations of the complexified symplectic structure David Favero VGIT and Derived Categories
Distribution of Calabi-Yau 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 ) . 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 ( � Fuk ( X ) X ) D b ( coh � D b ( coh X ) 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 → A 1 . David Favero VGIT and Derived Categories
A-model for an LG-model For a Landau-Ginzburg model w : X → A 1 . 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 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. 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 E 0 , and a pair of O X -module homomorphisms, : E 0 → E − 1 φ − 1 E E : E − 1 → E 0 φ 0 : E 0 → E 0 and E ◦ φ − 1 such that the compositions, φ 0 E E : E − 1 → E − 1 , are isomorphic to multiplication by w . φ − 1 ◦ φ 0 E 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 � Fuk ( X , w ) ( X , w ) D b ( coh � D b ( coh ( X , w )) ( 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, 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 semi-orthogonal decomposition by �A 1 , . . . , A m � . David Favero VGIT and Derived Categories
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 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 ( 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. David Favero VGIT and Derived Categories
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 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 ( 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. David Favero VGIT and Derived Categories
Mirror to Bl p P 2 > > 2 > ❍ ❍ ❍ ❍ ✉ ✉ ✉ ✉ ✟ ✟ ✟ ✟ 3 3 O O E i T (-1) O ( 1 ) 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 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. David Favero VGIT and Derived Categories
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. Blowing-down a point corresponds to deforming the K¨ ahler 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 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 y v P 2 Bl p ( P 2 ) x , u 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 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 semi-stable locus and X us ( 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 X ss ( L ) under the rational map � Γ( X , L n ) G . X ��� Proj n ≥ 0 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. 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 L 1 and L 2 such / L 1 C ∗ = X and Z / / L 2 C ∗ = Y that Z / 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, 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 G -equivariant line bundles with X ss � = ∅ . For each L ∈ Pic G ( X ) R , we have a cone C L = { L ′ ∈ Pic G ( X ) R : X L ⊂ X L ′ } . These are the cones of the fan. David Favero VGIT and Derived Categories
Blow-up - Blow-down We can realize Bl p ( P 2 ) as a GIT quotient of A 4 by the subgroup m ∼ = { ( r , r − 1 s , r , s ) : r , s ∈ G m } ⊂ G 4 G 2 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 . G -equivariant structures on the trivial bundle amount to characters of G 2 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 y v P 2 Bl p ( P 2 ) x , u David Favero VGIT and Derived Categories
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