Morse decomposition for D-module categories on stacks


McGerty, K
Nevins, T

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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.

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Journal Article