Categorical cell decomposition of quantized symplectic algebraic varieties

Abstract

We prove a new symplectic analogue of Kashiwara's equivalence from D -module theory. As a consequence, we establish a structure theory for module categories over deformation-quantizations that mirrors, at a higher categorical level, the Białynicki-Birula stratification of a variety with an action of the multiplicative group G m . The resulting categorical cell decomposition provides an algebrogeometric parallel to the structure of Fukaya categories of Weinstein manifolds. From it, we derive concrete consequences for invariants such as K -theory and Hochschild homology of module categories of interest in geometric representation theory. 53D55; 14F05

Very stable Higgs bundles, equivariant multiplicity and mirror symmetry

“…For this we need Lemma 6. 7 We can choose the étale cover i U i → M s so that the U i are Tequivariant and the local universal Higgs bundle (E U i , U i ) on U i ×C carries a T-equivariant structure with T acting on U i with weight −1. Furthermore the isomorphism (6.14) is T-equivariant with some T-equivariant structure on L.…”

“…There seems to be a related confusion in[7, Definition 1.1]. Their definition of elliptic Taction should either require the existence of limit points in the t → 0 limit as in this paper, or replace the definition of positive weight using (m t ) * instead of m * t .…”

“…Their definition of elliptic Taction should either require the existence of limit points in the t → 0 limit as in this paper, or replace the definition of positive weight using (m t ) * instead of m * t . In fact, when they use the definition of positive weight in the displayed line after (2.1) on page 2613 of[7], they use the correct (m t ) * instead of m * t as in [7, Definition 1.1].…”

Enumerative geometry and geometric representation theory

“…5 In K-theory, there is a correction to the gluing formula as in the work of Givental [50]. 6 This does not trivialize DT counts, in particular, does not make them factor through algebraic cobordism, for the following reasons.…”

“…Quantization of equivariant symplectic resolutions is a very fertile ground which is currently being explored by several teams of researchers, see for example [3,6,10,11,12,13,14,15,19,21,63,70,71,72,78]. A quantization of X is, first, a sheaf O p X of noncommutative algebras deforming the sheaf pO X , t¨,¨uq of Poisson algebras and, second, the algebra X " Γ`O p Xȏ f its global sections 12 .…”