[Paper Review] McKay equivalence for symplectic resolutions of singularities

Roman Bezrukavnikov, D. Kaledin|arXiv (Cornell University)|Jan 1, 2004

Algebraic Geometry and Number TheoryMathematics16 references88 citations

TL;DR

This paper establishes a derived McKay equivalence between the bounded derived category of coherent sheaves on a symplectic resolution $X$ of a quotient singularity $V/\Gamma$ and the derived category of $\Gamma$-equivariant coherent sheaves on $V$, using quantization and reduction to positive characteristic. The key result is an equivalence $D^b(\operatorname{Coh}(X)) \cong D^b(\operatorname{Coh}^\Gamma(V))$, extending earlier results to higher dimensions via Azumaya algebras and Brauer group techniques.

ABSTRACT

Let $V$ be a finite-dimensional symplectic vector space over a field of characteristic 0, and let $G \subset Sp(V)$ be a finite subgroup. We prove that for any crepant resolution $X o V/G$, the bounded derived category $D^b(Coh(X))$ of coherent sheaves on $X$ is equivalent to the bounded derived category $D^b_G(Coh(V))$ of $G$-equivariant coherent sheaves on $V$.

Motivation & Objective

• To generalize the McKay correspondence to higher-dimensional symplectic quotient singularities using derived categories.
• To establish a derived equivalence between the derived category of coherent sheaves on a symplectic resolution $X$ and the derived category of $\Gamma$-equivariant coherent sheaves on $V$.
• To provide a new proof of the $n!$-conjecture for Hilbert schemes of points on $\mathbb{A}^2$ via moduli interpretation.
• To show that any crepant resolution of a symplectic quotient singularity is a moduli space of $\Gamma$-constellations under suitable stability conditions.

Proposed method

• Reduction to positive characteristic via base change to a field $\mathsf{k}$ of large characteristic $p>0$.
• Construction of a quantization $\mathcal{O}_h(X)$ of the structure sheaf $\mathcal{O}_X$ as a deformation over $\mathsf{k}[[h]]$ with global sections isomorphic to the $\Gamma$-invariant Weyl algebra $\mathcal{W}^\Gamma$.
• Identification of the quantized algebra $\mathcal{O}_h$ as an Azumaya algebra over the Frobenius twist $X^{(1)}$, leading to a gerbe structure.
• Use of the norm map on Brauer groups to untwist the Azumaya algebra and recover a derived equivalence over $\mathsf{k}$.
• Application of Morita equivalence between $\mathcal{W}^\Gamma$ and $\mathcal{W}\#\Gamma$ in large characteristic to relate the derived categories.
• Lifting the equivalence from positive characteristic to characteristic zero using a limit argument and base change.

Experimental results

Research questions

• RQ1 Does a derived McKay equivalence hold for symplectic resolutions of quotient singularities in arbitrary dimension?
• RQ2 Can the equivalence between $D^b(\operatorname{Coh}(X))$ and $D^b(\operatorname{Coh}^\Gamma(V))$ be established beyond the 2D case using quantization?
• RQ3 Is the resolution $X$ of $V/\Gamma$ a moduli space of $\Gamma$-equivariant sheaves (G-constellations) with appropriate stability conditions?
• RQ4 Can the $n!$-conjecture for Hilbert schemes of points on $\mathbb{A}^2$ be proven without explicit computation via this equivalence?

Key findings

• The derived category $D^b(\operatorname{Coh}(X))$ is equivalent to $D^b(\operatorname{Coh}^\Gamma(V))$ for any symplectic resolution $X \to V/\Gamma$, generalizing the 2D case.
• The quantization $\mathcal{O}_h(X)$ has global sections isomorphic to the $\Gamma$-invariant Weyl algebra $\mathcal{W}^\Gamma$, and is an Azumaya algebra over $X^{(1)}$ in positive characteristic.
• The global sections of the Azumaya algebra on $X^{(1)}$ are Morita equivalent to $\mathcal{W}\#\Gamma$, enabling a derived equivalence over $\mathsf{k}$.
• The equivalence over $\mathsf{k}$ of large positive characteristic lifts to characteristic zero via base change and completion, proving Theorem 1.1.
• The resolution $X$ is shown to be a moduli space of $\Gamma$-constellations, though precise stability conditions are left for future work.
• The equivalence implies $H^p(X, \Omega^q_X) = 0$ for $p > q$, a consequence of Hochschild homology invariance under derived equivalences.