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 Motivating Example 1 Background on GIT 2 General results 3 Landau-Ginzburg models and factorizations 4 RG-flow and a theorem of Orlov 5 David Favero VGIT and Derived Categories
Motivating Example Outline Motivating Example 1 Background on GIT 2 General results 3 Landau-Ginzburg models and factorizations 4 RG-flow and a theorem of Orlov 5 David Favero VGIT and Derived Categories
Motivating Example Weighted Projective Stacks Consider k [ x 0 , x 1 , x 2 ] with the G m -action with weights ( 1 , 1 , n ) . We define P ( 1 : 1 : n ) as the smooth global quotient Deligne-Mumford stack, [( Spec k [ x 0 , x 1 , x 2 ] \ 0 ) / G m ] . Characters of G m , λ �→ λ i , give line bundles, O ( i ) , and a tilting object, T , is given by, n + 2 � T := O ( i ) i = 0 Quiver for P ( 1 : 1 : 4 ) : x 2 x 0 x 0 x 0 x 0 x 0 • • • • • • x 1 x 1 x 1 x 1 x 1 x 2 David Favero VGIT and Derived Categories
Motivating Example Hirzebruch Surfaces Consider the total space of O P 1 ⊕ O P 1 ( − n ) with the G m -action given by dilating the fibers. The Hirzebruch surface, F n , is defined as the projective bundle, P ( O P 1 ⊕ O P 1 ( − n )) , which represents the smooth global quotient stack, [ tot ( O P 1 ⊕ O P 1 ( − n )) \ zero section / G m ] . A tilting object, T , is given by, T := O ⊕ π ∗ O ( 1 ) ⊕ O π ( 1 ) ⊕ π ∗ O ( 1 ) ⊗ O π ( 1 ) . x 0 • • x 1 x 3 0 x 2 0 x 1 x 2 x 2 x 0 x 2 1 x 3 1 x 0 • • x 1 David Favero VGIT and Derived Categories
Motivating Example Comparing Quiver for P ( 1 : 1 : 4 ) : x 2 x 0 x 0 x 0 x 0 x 0 • • • • • • x 1 x 1 x 1 x 1 x 1 x 2 Quiver for F n : x 0 • • x 1 x 3 0 x 2 0 x 1 x 2 x 2 x 0 x 2 1 x 3 1 x 0 • • x 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, 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
Motivating Example Hirzebruch surfaces • If n < 2, there is a semi-orthogonal decomposition D b ( coh F n ) = � E 1 , . . . , E 2 − n , D b ( coh P ( 1 , 1 , n )) � with E i exceptional objects. • If n = 2, we have an equivalence D b ( coh F n ) = D b ( coh P ( 1 , 1 , 2 )) . • If n > 2, there is a semi-orthogonal decomposition D b ( coh P ( 1 , 1 , n )) = � E 1 , . . . , E n − 2 , D b ( coh F n ) � with E i exceptional objects. David Favero VGIT and Derived Categories
Background on GIT Outline Motivating Example 1 Background on GIT 2 General results 3 Landau-Ginzburg models and factorizations 4 RG-flow and a theorem of Orlov 5 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, X ss ( L ) := { x ∈ X | ∃ f ∈ H 0 ( X , L n ) G with n ≥ 0 , f ( x ) � = 0 , and X f affine } For us, the GIT quotient corresponding to this data is the global quotient stack [ X ss ( L ) / G ] . We can vary the G -equivariant structure on L by choosing characters, χ , in the dual group, � G := Hom ( G , G m ) . 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, A χ , is the complement of the semistable locus in X . Let X be proper or affine. There exists a fan in � G R with support the G with X ss � = ∅ . For each χ ∈ � set of characters in � G , we have a cone C χ = { µ ∈ � G R : A µ ⊂ A χ } . These are the cones of the fan. The characters on the relative interiors of 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 F n as a GIT quotient of A 4 by the subgroup { ( r , r − n 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 . David Favero VGIT and Derived Categories
Background on GIT Hirzebruch surfaces The GIT fan for this quotient is y v P ( 1 , 1 , n ) F n x , u 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 : G m → 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 ( G m ) -action and an induced linearization (pulling back L ). Let Z λ denote the semi-stable locus and let S ± λ be S + λ := { x ∈ X | lim t →∞ λ ( t ) · x ∈ Z λ } S − λ := { x ∈ X | lim t → 0 λ ( t ) · x ∈ Z λ } . 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, λ i : G m → G ,with X us ( L ) = G · S + λ 1 ∪ · · · ∪ G · S + λ p . David Favero VGIT and Derived Categories
General results Outline Motivating Example 1 Background on GIT 2 General results 3 Landau-Ginzburg models and factorizations 4 RG-flow and a theorem of Orlov 5 David Favero VGIT and Derived Categories
General results Setup Suppose that we have a one-parameter family of linearizations, L t , such that X ss ( L 0 ) = G · S + λ 1 ∪ · · · ∪ G · S + λ p ∪ X ss ( L t ) for t > 0 X ss ( L 0 ) = G · S − λ 1 ∪ · · · ∪ G · S − λ p ∪ X ss ( L t ) 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 / − . If p = 1, let Y be the GIT quotient associated to Z λ 1 . t < 0 as X / Choose a fixed point x ∈ Z λ i . Let µ i be the sum of the weights of G m -action on the normal bundles to G · S + λ i and G · S − λ i restricted to x . David Favero VGIT and Derived Categories
General results Main theorem Theorem (Ballard-F-Katzarkov, Halpern-Leinster) Fix d 1 , . . . , d p ∈ Z . If µ i > 0 for all 1 ≤ i ≤ p , then there exists a left-admissible fully-faithful functor, Φ d 1 ,..., d p : D b ( coh X / / − ) → D b ( coh X / / +) . If p = 1, then there also exists fully-faithful functors, Υ − j : D b ( coh Y ) → D b ( coh X / / +) , and a semi-orthogonal decomposition, / +) = � Υ − D b ( coh X / − d D b ( coh Y ) , . . . , Υ − µ − d − 1 D b ( coh Y ) , Φ d D b ( coh X / / − ) � . David Favero VGIT and Derived Categories
General results Main theorem Theorem (Ballard-F-Katzarkov, Halpern-Leinster) Fix d 1 , . . . , d p ∈ Z . If µ i = 0 for all 1 ≤ i ≤ p , then there exist an equivalence, Φ d 1 ,..., d p : D b ( coh X / / − ) → D b ( coh X / / +) . David Favero VGIT and Derived Categories
General results Main theorem Theorem (Ballard-F-Katzarkov, Halpern-Leinster) Fix d 1 , . . . , d p ∈ Z . If µ < 0 for all 1 ≤ i ≤ p , then there exists a left-admissible fully-faithful functor, Ψ d 1 ,..., d p : D b ( coh X / / +) → D b ( coh X / / − ) If p = 1, then there also exists fully-faithful functors, Υ + j : D b ( coh Y ) → D b ( coh X / / +) , and a semi-orthogonal decomposition, / − ) = � Υ + D b ( coh X / − d D b ( coh Y ) , . . . , Υ + µ − d + 1 D b ( coh Y ) , Ψ d D b ( coh X / / +) � . David Favero VGIT and Derived Categories
General results Simplifications If the stabilizer of x is G m , then µ i > 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, − K X � . David Favero VGIT and Derived Categories
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