Categorical cell decomposition of quantized symplectic algebraic varieties

Bellamy, Gwyn¹ ; Dodd, Christopher² ; McGerty, Kevin³ ; Nevins, Thomas²

¹ School of Mathematics and Statistics, University of Glasgow, 15 University Gardens, Glasgow, G12 8QW, United Kingdom
² Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W Green Street, MC-382, Urbana, IL 61801, United States
³ Mathematical Institute, University of Oxford, Andrew Wiles Building, Woodstock Road, Oxford, OX2 6GG, United Kingdom

Geometry & topology, Tome 21 (2017) no. 5, pp. 2601-2681.

Résumé

We prove a new symplectic analogue of Kashiwara’s equivalence from $\mathcal{D}$–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 ${\mathbb{G}}_m$. 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.

Détail

DOI : 10.2140/gt.2017.21.2601

Classification : 53D55, 14F05
Keywords: quantization, symplectic, elliptic

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