Morse decomposition for D-module categories on stacks

TL;DR

This paper develops a Morse decomposition framework for D-module categories on stacks, extending functorial properties and providing new insights into the topology of symplectic quotients.

Contribution

It introduces a recollement package for twisted D-modules on stacks with Kirwan-Ness stratification, extending standard functorialities and relating to Morse theory and Kirwan surjectivity.

Findings

Established a recollement package for twisted D-modules on stacks.

Extended functoriality properties for categories of twisted D-modules.

Derived a modified Kirwan surjectivity result for hyperkähler quotients.

Abstract

Let Y be a smooth algebraic stack exhausted by quotient stacks. Given a Kirwan-Ness stratification of the cotangent stack T^*Y, we establish a recollement package for twisted D-modules on Y, gluing the category from subquotients described via modules microsupported on the Kirwan-Ness strata of T^*Y. The package includes unusual existence and "preservation-of-finiteness'' properties for functors of the full category of twisted D-modules, extending the standard functorialities for holonomic modules. In the case that Y = X/G is a quotient stack, our results provide a higher categorical analogue of the Atiyah-Bott--Kirwan--Ness "equivariant perfection of Morse theory'' for the norm-squared of a real moment map. As a consequence, we deduce a modified form of Kirwan surjectivity for the cohomology of hyperkaehler/algebraic symplectic quotients of cotangent bundles.