Dorin Boger | Searcharxiv

arXiv2016

A Parameterization of D equivalences of coherent sheaves of symplectic resolutions of a given symplectic singularity

Let G be a reductive groups over an algebraically closed field k. Let P^{(i)} be associated parabolic subgroups, and X^{(i)}:=T^*G/P^i. The bounded derived categories of coherent sheaves on X^{(i)} are equivalent, but there is no canonical equivalence. By refining a construction from a previous paper, we construct a local system of categories over a topological space V^0_C, where these categories are assigned to different points in V^0_C. Natural equivalence functors between these categories are parameterized by homotopy classes of paths between the corresponding points.

math.AG↗

arXiv2016

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

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 $π_1 V^0_{\mathbb{C}}$ 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.

math.AG↗