Groups actions, and D equivalences of categories of coherent sheaves of symplectic resolutions

TL;DR

This paper constructs a group action on the derived category of coherent sheaves for symplectic resolutions in positive characteristic and explores equivalences between such categories for different resolutions, with implications for characteristic zero.

Contribution

It introduces a local system on a topological space that induces a group action on derived categories of coherent sheaves for symplectic resolutions.

Findings

Constructed a local system with values in derived categories of cotangent bundles of G/P.

Induced an action of the fundamental group on the derived category.

Connected equivalence functors to homotopy classes of maps in the base space.

Abstract

Let k be an algebraically closed field of characteristic p>>0. Let $X\rightarrow Y$ be a symplectic resolution. There are two questions which motivates this work. One question is a construction of an action of a group on the category $\mathcal{C}:=D^b(Coh(X))$ - The bounded derived category of coherent sheaves of the symplectic resolution X. Second question is understanding equivalence functors between derived categories of coherent sheaves for different symplectic resolutions of Y. Let G/k be a reductive group. In this paper, we construct a local system on a topological space called $V^0_{\mathbb{C}}$ with value the category $D^b(Coh(T^*G/P))$ for a parabolic subgroup P. This induces an action of $\pi_1 V^0_{\mathbb{C}}$ on the category. In another paper we further explain how a refinement of this local system construction, gives an answer to the second question, showing that these equivalence functors, are parametrized by homotopy classes of maps between certain points in the base space. We also lift the result to characteristic zero.