Stability conditions for noncommutative symplectic resolutions - PowerPoint PPT Presentation

Stability conditions for noncommutative symplectic resolutions Gufang Zhao Northeastern University Conference on Geometric Methods in Representation Theory 2013 Gufang Zhao (Northeastern) Noncommutative symplectic resolutions Nov. 25, 2013 Columbia, Mo 1 / 22

Outline Motivation 1 The dimension polynomials 2 Real variation of stability conditions 3 The t -structures associated to alcoves 4 Gufang Zhao (Northeastern) Noncommutative symplectic resolutions Nov. 25, 2013 Columbia, Mo 2 / 22

Motivation Symplectic resolutions: commutative vs. non-comm. We work over a separably closed field k of characteristic p >> 0. Let Γ n := ( Z r ) n ⋊ S n acting on h = A n in the natural way. Let V = h ⊕ h ∗ � A 2 n be endowed with the diagonal action of Γ n . The action preserves the natural symplectic form on V . A symplectic resolution of A 2 n / Γ n can be given as Hilb n ( � A 2 / Z r ) where � A 2 / Z r is the minimal resolution of A 2 / Z r . Let W ( h ) be the Weyl algebra. The algebra W ( h ) Γ n is a noncommutative desingularization of A 2 n / Γ n . Gufang Zhao (Northeastern) Noncommutative symplectic resolutions Nov. 25, 2013 Columbia, Mo 3 / 22

Motivation The rational Cherednik algebras The (spherical) Cherednik algebra s H c is a deformation of W ( h ) Γ n . The parameter space of the deformation is spanned by the conjugacy classes of reflections in Γ n . The algebra s H c has a big Frobenius center k [ A 2 n ( 1 ) ] Γ n . For any central character χ , the irreducible objects in the category Mod- χ H c are naturally labeled by Irrep (Γ n ) . If s H c has finite global dimension, then the value c is called spherical value . Otherwise we say c is aspherical . The aspherical values form an affine hyperplane arrangement in the space of parameters. Gufang Zhao (Northeastern) Noncommutative symplectic resolutions Nov. 25, 2013 Columbia, Mo 4 / 22

Motivation The Localization Theorem Let Hilb ( 1 ) be the Frobenius twist of Hilb := Hilb n ( � A 2 / Z r ) . Let Coh 0 Hilb ( 1 ) be the category of coherent sheaves on Hilb ( 1 ) set-theoretically supported on the zero-fiber of the Hilbert-Chow morphism. Theorem (Bezrukavnikov-Finkelberg-Ginzburg) There is a tilting bundle E c on Hilb ( 1 ) , such that End ( E c ) ˆ 0 � ( s H c ) ˆ 0 In particular, for spherical values c, there is a derived equivalence D b ( Coh 0 Hilb ( 1 ) ) � D b ( Mod - 0 s H c ) . Gufang Zhao (Northeastern) Noncommutative symplectic resolutions Nov. 25, 2013 Columbia, Mo 5 / 22

Motivation The t -structures For each spherical value c , the derived equivalence D b ( Coh 0 Hilb ( 1 ) ) � D b ( Mod- 0 s H c ) endows D b ( Coh 0 Hilb ( 1 ) ) with a t -structure. Question For two different spherical values c and c ′ , what is the relation between the t-structures on D b ( Coh 0 Hilb ( 1 ) ) ? If c and c ′ are in the same alcove, then the translation functor induces a Morita equivalence; the t -structures are the same. Gufang Zhao (Northeastern) Noncommutative symplectic resolutions Nov. 25, 2013 Columbia, Mo 6 / 22

The dimension polynomials Table of Contents Motivation 1 The dimension polynomials 2 Real variation of stability conditions 3 The t -structures associated to alcoves 4 Gufang Zhao (Northeastern) Noncommutative symplectic resolutions Nov. 25, 2013 Columbia, Mo 7 / 22

The dimension polynomials Dimensions of irreducible modules Let L c ( τ ) be the irreducible object in Mod- 0 s H c labeled by τ ∈ Irrep (Γ n ) . Under the BFG-derived equivalence, Mod- 0 s H c ∋ L c ( τ ) ↔ L c ( τ ) ∈ D b ( Coh 0 Hilb ( 1 ) ) ; Mod- End ( E c ) ∋ proj. cover of L c ( τ ) ↔ V τ vector bundle on Hilb. When c + ν is in the same alcove as c , dim L c + ν ( τ ) := χ ( L c ( τ ) ⊗ E 0 ⊗ O ( ν )) is a polynomial in ν . Proposition For a basis { ch O ( b ) } of H ∗ ( Hilb , Q l ) with b = � h b τ ch ( V τ ) , we have χ ( L τ ⊗ O ( b )) = h b τ . Gufang Zhao (Northeastern) Noncommutative symplectic resolutions Nov. 25, 2013 Columbia, Mo 8 / 22

The dimension polynomials The Chern character problem Theorem (Etingof-Ginzburg, Ginzburg-Kaledin) The algebra H ∗ ( Hilb ; Q ) is isomorphic to the algebra gr Z Q [Γ n ] . The following problem is raised by Etingof, Ginzburg, and Kaledin. Problem Express explicitly the map K 0 (Γ n ) → gr Z Q [Γ n ] induced by the Chern character ch : K 0 ( Hilb ) → H ∗ ( Hilb ; Q ) . Gufang Zhao (Northeastern) Noncommutative symplectic resolutions Nov. 25, 2013 Columbia, Mo 9 / 22

� The dimension polynomials The Chern character maps: an example Take Γ = Z 2 acting on A 2 . The minimal resolution is T ∗ P 1 , the central fiber is the zero section P 1 . The quiver is the affine A 1 quiver with v = ( 1 , 1 ) and w = ( 1 , 0 ) . X 1 � v 1 v 0 X 2 There is a natural G m -action with fixed points [ 1 , 0 ] and [ 0 , 1 ] . At [ 1 , 0 ] , v 1 has weight 1; at [ 0 , 1 ] , v 1 has weight -1. Therefore, equivariant localization theorem tells us that c 1 ( V 2 ) = − 1. Gufang Zhao (Northeastern) Noncommutative symplectic resolutions Nov. 25, 2013 Columbia, Mo 10 / 22

The dimension polynomials Dimension polynomials via quasi-invariants For an integral parameter m , let Q m be the m -quasi-invariants in k [ h ] . As Γ n - s H m bimodule, Q m = ⊕ τ ∈ Irrep (Γ n ) τ ∗ ⊗ M m ( τ ) . Let � Q m be the quasi-invariants on the Frobenius neighborhood of 0. Q m : · · · → Q m ⊗ ∧ 2 h ( 1 ) → Q m ⊗ h ( 1 ) → Q m . A resolution of � Theorem (Z., to appear) Fix a character i of Z r . Let τ ( i ) be the 1-dimensional representation of Γ n = ( Z r ) n ⋊ S n on which Z r acts by the character i and S n acts by the sign representation. The Poincaré series of L m ( τ ( i )) is t ni � n − 1 k = 0 ( 1 − t rk + m 0 n + p + 1 + rm i + 1 ) � n . k = 1 ( 1 − t kr ) Gufang Zhao (Northeastern) Noncommutative symplectic resolutions Nov. 25, 2013 Columbia, Mo 11 / 22

Real variation of stability conditions Table of Contents Motivation 1 The dimension polynomials 2 Real variation of stability conditions 3 The t -structures associated to alcoves 4 Gufang Zhao (Northeastern) Noncommutative symplectic resolutions Nov. 25, 2013 Columbia, Mo 12 / 22

Real variation of stability conditions The Central Charge We reparameterize the value c by setting x = cp and define p →∞ p − n dim k L c ( τ ; p ) . Z τ ( x ) = lim We consider the collection of polynomials { Z τ ( x ) | τ ∈ Irrep (Γ n ) } as a polynomial map H 2 ( Hilb ; R ) → Hom Z ( K 0 ( Hilb ) , R ) . This polynomial map is called the central charge . Gufang Zhao (Northeastern) Noncommutative symplectic resolutions Nov. 25, 2013 Columbia, Mo 13 / 22

Real variation of stability conditions The Main Theorem Assume n = 2. T : { alcoves } → { t -structures } ; A �→ Mod 0 s H c (Γ 2 ) ⊆ D b ( Coh 0 ( Hilb )) for c ∈ A . Z : H 2 ( Hilb ; R ) → Hom Z ( K 0 ( Hilb ) , R ) : the central charge. Theorem (Z., to appear) The pair ( T , Z ) is a real variation of stability conditions. More concretely, for any alcove A, let A := heart of T ( A ) . We have, for any x ∈ A, Z L ( x ) > 0 for any simple object L ∈ A ; 1 for any A ′ , sharing a codim. 1 wall H with A. 2 Let A � A i := � L ∈ A | Z L ( x ) vanishes of order ≥ i on H � . Then, ◮ the T ( A ′ ) is compatible with the filtration on T ( A ) ; ◮ on gr i ( A ) = A i / A i + 1 , φ ( A ′ ) differs by [ i ] from φ ( A ) . Gufang Zhao (Northeastern) Noncommutative symplectic resolutions Nov. 25, 2013 Columbia, Mo 14 / 22

The t -structures associated to alcoves Table of Contents Motivation 1 The dimension polynomials 2 Real variation of stability conditions 3 The t -structures associated to alcoves 4 Gufang Zhao (Northeastern) Noncommutative symplectic resolutions Nov. 25, 2013 Columbia, Mo 15 / 22

The t -structures associated to alcoves The hyperplane arrangement for Γ 1 = Z r and n = 2 The t -structure t 0 is generated by the Procesi bundle on Hilb. t 1 is obtained from t 0 by a P 2 -semi-reflection. b t 2 t 1 t 0 a Gufang Zhao (Northeastern) Noncommutative symplectic resolutions Nov. 25, 2013 Columbia, Mo 16 / 22

The t -structures associated to alcoves The P n -semi-reflections A = an abelian category, of finite length, having finitely many simples; either A has enough projective objects; or A = Mod- χ H , for some algebra H , finitely generated over its center Z ( H ) , and a central character χ . R S A := the (right) tilting of A with respect to a simple S . Let S θ be a simple object with Ext 1 ( S θ , S θ ) = 0. Proposition Assume the projective objects in R S θ [ n − 1 ] R S θ [ n − 2 ] · · · R S θ ( A ) are concentrated in degree zero. Then for all 0 ≤ i ≤ n, R S θ [ i − 1 ] R S θ [ i − 2 ] · · · R S θ ( A ) has a set of projectives consisting of objects lying in A , is derived equivalent to A . Gufang Zhao (Northeastern) Noncommutative symplectic resolutions Nov. 25, 2013 Columbia, Mo 17 / 22

The t -structures associated to alcoves An example of P n -semi-reflection Endow P n with the standard Bruhat stratification. A = { perverse sheaves on P n } . S n := C P n [ n ] , simple object in A . It is an P n object. The semi-reflection of A with respect to S n can be obtained by iterated tilting. The projective generators consist of perverse sheaves. Gufang Zhao (Northeastern) Noncommutative symplectic resolutions Nov. 25, 2013 Columbia, Mo 18 / 22