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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

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Stability conditions for noncommutative symplectic resolutions

Gufang Zhao

Northeastern University

Conference on Geometric Methods in Representation Theory 2013

Gufang Zhao (Northeastern) Noncommutative symplectic resolutions

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Outline

1

Motivation

2

The dimension polynomials

3

Real variation of stability conditions

4

The t-structures associated to alcoves

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Motivation

Symplectic resolutions: commutative vs. non-comm.

We work over a separably closed field k of characteristic p >> 0. Let Γn := (Zr)n ⋊ Sn acting on h = An in the natural way. Let V = h ⊕ h∗ A2n be endowed with the diagonal action of Γn. The action preserves the natural symplectic form on V. A symplectic resolution of A2n/Γn can be given as Hilbn(

A2/Zr)

where

A2/Zr is the minimal resolution of A2/Zr.

Let W(h) be the Weyl algebra. The algebra W(h)Γn is a noncommutative desingularization of A2n/Γn.

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Motivation

The rational Cherednik algebras

The (spherical) Cherednik algebra sHc is a deformation of W(h)Γn. The parameter space of the deformation is spanned by the conjugacy classes of reflections in Γn. The algebra sHc has a big Frobenius center k[A2n(1)]Γn. For any central character χ, the irreducible objects in the category Mod-χ Hc are naturally labeled by Irrep(Γn). If sHc 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.

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Motivation

The Localization Theorem

Let Hilb(1) be the Frobenius twist of Hilb := Hilbn(

A2/Zr).

Let Coh0 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 Ec on Hilb(1), such that End(Ec)ˆ

0 (sHc)ˆ

In particular, for spherical values c, there is a derived equivalence Db(Coh0 Hilb(1)) Db(Mod-0

sHc).

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Motivation

The t-structures

For each spherical value c, the derived equivalence Db(Coh0 Hilb(1)) Db(Mod-0

sHc)

endows Db(Coh0 Hilb(1)) with a t-structure.

Question

For two different spherical values c and c′, what is the relation between the t-structures on Db(Coh0 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.

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The dimension polynomials

Table of Contents

1

Motivation

2

The dimension polynomials

3

Real variation of stability conditions

4

The t-structures associated to alcoves

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The dimension polynomials

Dimensions of irreducible modules

Let Lc(τ) be the irreducible object in Mod-0 sHc labeled by τ ∈ Irrep(Γn). Under the BFG-derived equivalence, Mod-0 sHc ∋ Lc(τ) ↔ Lc(τ) ∈ Db(Coh0 Hilb(1)); Mod- End(Ec) ∋ proj. cover of Lc(τ) ↔ Vτ vector bundle on Hilb. When c + ν is in the same alcove as c, dim Lc+ν(τ) := χ(Lc(τ) ⊗ E0 ⊗ O(ν)) is a polynomial in ν.

Proposition

For a basis {ch O(b)} of H∗(Hilb, Ql) with b = hb

τ ch(Vτ), we have

χ(Lτ ⊗ O(b)) = hb

τ .

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The dimension polynomials

The Chern character problem

Theorem (Etingof-Ginzburg, Ginzburg-Kaledin)

The algebra H∗(Hilb; Q) is isomorphic to the algebra gr ZQ[Γn]. The following problem is raised by Etingof, Ginzburg, and Kaledin.

Problem

Express explicitly the map K0(Γn) → gr ZQ[Γn] induced by the Chern character ch : K0(Hilb) → H∗(Hilb; Q).

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The dimension polynomials

The Chern character maps: an example

Take Γ = Z2 acting on A2. The minimal resolution is T∗P1, the central fiber is the zero section P1. The quiver is the affine A1 quiver with v = (1, 1) and w = (1, 0). v0

X1

v1

X2

  • There is a natural Gm-action with fixed points [1, 0] and [0, 1].

At [1, 0], v1 has weight 1; at [0, 1], v1 has weight -1. Therefore, equivariant localization theorem tells us that c1(V2) = −1.

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The dimension polynomials

Dimension polynomials via quasi-invariants

For an integral parameter m, let Qm be the m-quasi-invariants in k[h]. As Γn-sHm bimodule, Qm = ⊕τ∈Irrep(Γn)τ∗ ⊗ Mm(τ). Let Qm be the quasi-invariants on the Frobenius neighborhood of 0. A resolution of Qm: · · · → Qm ⊗ ∧2h(1) → Qm ⊗ h(1) → Qm.

Theorem (Z., to appear)

Fix a character i of Zr. Let τ(i) be the 1-dimensional representation of

Γn = (Zr)n ⋊ Sn on which Zr acts by the character i and Sn acts by the

sign representation. The Poincaré series of Lm(τ(i)) is tni n−1

k=0(1 − trk+m0n+p+1+rmi+1)

n

k=1(1 − tkr)

.

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Real variation of stability conditions

Table of Contents

1

Motivation

2

The dimension polynomials

3

Real variation of stability conditions

4

The t-structures associated to alcoves

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Real variation of stability conditions

The Central Charge

We reparameterize the value c by setting x = cp and define Zτ(x) = lim

p→∞ p−n dimk Lc(τ; p).

We consider the collection of polynomials {Zτ(x) | τ ∈ Irrep(Γn)} as a polynomial map H2(Hilb; R) → HomZ(K0(Hilb), R). This polynomial map is called the central charge.

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Real variation of stability conditions

The Main Theorem

Assume n = 2. T : {alcoves} → {t-structures}; A → Mod0sHc(Γ2) ⊆ Db(Coh0(Hilb)) for c ∈ A. Z : H2(Hilb; R) → HomZ(K0(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,

1

for any x ∈ A, ZL(x) > 0 for any simple object L ∈ A;

2

for any A′, sharing a codim. 1 wall H with A. Let A Ai := L ∈ A | ZL(x) vanishes of order ≥ i on H. Then,

◮ the T(A′) is compatible with the filtration on T(A); ◮ on gri(A) = Ai/Ai+1, φ(A′) differs by [i] from φ(A). Gufang Zhao (Northeastern) Noncommutative symplectic resolutions

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The t-structures associated to alcoves

Table of Contents

1

Motivation

2

The dimension polynomials

3

Real variation of stability conditions

4

The t-structures associated to alcoves

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The t-structures associated to alcoves

The hyperplane arrangement for Γ1 = Zr and n = 2

The t-structure t0 is generated by the Procesi bundle on Hilb. t1 is obtained from t0 by a P2-semi-reflection.

t1 t0 t2 b a

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The t-structures associated to alcoves

The Pn-semi-reflections

A = an abelian category, of finite length, having finitely many simples;

either A has enough projective objects;

  • r A = Mod-χ H, for some algebra H, finitely generated over its

center Z(H), and a central character χ. RSA := the (right) tilting of A with respect to a simple S. Let Sθ be a simple object with Ext1(Sθ, Sθ) = 0.

Proposition

Assume the projective objects in RSθ[n−1]RSθ[n−2] · · · RSθ(A) are concentrated in degree zero. Then for all 0 ≤ i ≤ n, RSθ[i−1]RSθ[i−2] · · · RSθ(A) has a set of projectives consisting of objects lying in A, is derived equivalent to A.

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The t-structures associated to alcoves

An example of Pn-semi-reflection

Endow Pn with the standard Bruhat stratification.

A = {perverse sheaves on Pn}.

Sn := CPn[n], simple object in A. It is an Pn object. The semi-reflection of A with respect to Sn can be obtained by iterated tilting. The projective generators consist of perverse sheaves.

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The t-structures associated to alcoves

Example: Perverse sheaves on Pn

For simplicity, we take n = 2. The category A is Morita equivalent to

  • pt

α

  • A1

δ

  • β
  • A2

γ

  • with relations αβ = 0, δγ = 0, δα = 0, and βγ = 0. The projective objects:

Ppt = C2

pt α

  • CA1

β

  • ;

PA1 = Cpt

C2

A1 β

  • δ
  • CA2

γ

  • ;

PA2 = CA1

CA2

γ

  • .

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The t-structures associated to alcoves

Example: Perverse sheaves on Pn

Tilting with respect to S2: the tilting generators: Ppt, PA1, and coker(PA2 → PA1) P′

A2 = Cpt

CA1

β

  • .

Tilting with respect to S2[1]: the tilting generators: Ppt, PA1, and coker(P′

A2 → Ppt) P′′ A2 = Cpt.

The hearts of all these t-structures are derived equivalent to A.

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The t-structures associated to alcoves

Main ingredient: Lifting to characteristic zero

Let Z ⊆ R ⊂ C finitely generated over Z, such that sHc(Γ2)R exists. Let Oc be the category O of sHc(Γ2)C. Let c be aspherical, O≤d

c

:= Lc(τ; C) | codim supp Lc(τ; C) ≤ d;

And Mod-0 sHc(Γ2)≤d

k

:= Lc(τ; p) | deg(Zτ) ≤ d.

These two filtrations are compatible:

Theorem (Z., to appear)

Suppose the codimension of support of Lc(τ; C) is d. Lc(τ; C) Lc(τ; R)

  • Lc(τ; R) ⊗R k

central reduction

  • Lc(τ; R)k

Then Lc(τ; R)k is a nonzero irreducible object in Mod- sHc(Γ2)≤d

k / Mod- sHc(Γ2)≤d+1 k

.

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Thank You!!!