[Paper Review] Morse decomposition for D-module categories on stacks

Kevin McGerty, Thomas Nevins|arXiv (Cornell University)|Feb 28, 2014

Homotopy and Cohomology in Algebraic Topology18 references3 citations

TL;DR

This paper establishes a categorical Morse decomposition for twisted D-module categories on smooth algebraic stacks exhausted by quotient stacks, using Kirwan-Ness stratifications of the cotangent stack. It proves a recollement structure that glues D-module categories from microsupported subquotients on strata, with novel preservation-of-finiteness and existence properties for functors, leading to a modified Kirwan surjectivity for hyperkähler and algebraic symplectic quotients via Hochschild homology.

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.

Motivation & Objective

• To develop a categorical Morse decomposition for twisted D-modules on stacks using Kirwan-Ness stratifications of the cotangent stack.
• To establish a recollement structure that decomposes the derived category of D-modules into subquotients microsupported on KN strata.
• To prove new existence and finiteness properties for functors (e.g., $j_*$, $j^*$) in the unbounded derived category setting, extending holonomic functoriality.
• To deduce cohomological consequences for hyperkähler and algebraic symplectic quotients via Hochschild homology and the Kirwan map.
• To provide a higher categorical analogue of equivariant Morse theory for the norm-squared of a real moment map in the context of D-modules.

Proposed method

• Use a Kirwan-Ness stratification of the cotangent stack $T^*ancyscript{Y}$ for a smooth stack $ancyscript{Y}$ exhausted by quotient stacks $X/G$.
• Define microlocal categories ${ancyscript{D}}(ancyscript{U},c)$ as quotients of ${ancyscript{D}}(ancyscript{Y},c)$ by subcategories of objects microsupported on closed unions of strata.
• Construct a recollement via adjoint functors: $j^*: {ancyscript{D}}({ancyscript{U}},c) o {ancyscript{D}}({ancyscript{U}}^ullet,c)$, $j_*$, $i_*$, and $i^!$ for open and closed inclusions of strata.
• Prove that the right adjoint $j_*$ preserves compactness and coherence of D-modules, extending standard results from the holonomic case.
• Apply Hochschild homology to the derived categories, using the trace functor and duality to relate $HH_*({ancyscript{D}}({ancyscript{U}},c))$ to de Rham cohomology of symplectic quotients.
• Leverage the identification $HH_*({ancyscript{D}}(X/G)) o H^*_{dR}(I(X)/G)$ and the ${ancyscript{W}}$-algebra descent to relate Hochschild homology to cohomology of $ancyscript{Y}^{ss}/G$.

Experimental results

Research questions

• RQ1 How can one extend equivariant Morse theory to the setting of D-modules on stacks via a categorical decomposition?
• RQ2 What functorial properties (e.g., preservation of compactness and coherence) hold for pushforward and restriction functors in the unbounded derived category of twisted D-modules?
• RQ3 Can the Kirwan surjectivity theorem be generalized or modified in the context of hyperkähler and algebraic symplectic quotients using D-module categories?
• RQ4 To what extent does the Hochschild homology of D-module categories on stacks reflect the de Rham cohomology of symplectic quotients?
• RQ5 How does the microlocal structure of D-modules on stacks relate to the Kirwan-Ness stratification of the cotangent stack?

Key findings

• The paper establishes a recollement structure for twisted D-modules on stacks, decomposing the category via microsupported subquotients on Kirwan-Ness strata of $T^*ancyscript{Y}$.
• $j_*$, the right adjoint to restriction, preserves both compactness and coherence of D-modules, a non-trivial extension beyond the holonomic case.
• The restriction functor $j^*$ induces a surjection on Hochschild homology, as shown via the composition $j^* o j_!$ being a homotopy inverse.
• The Hochschild homology of ${ancyscript{D}}(X/G)$ is isomorphic to the de Rham cohomology of the inertia stack $I(X)/G$, with a shift: $HH_k o H^*_{dR}^{2n-k}$.
• The derived category ${ancyscript{D}}(T^*X^{ss}/G)$ is equivalent to the derived category of modules over a ${ancyscript{W}}$-algebra on $ancyscript{Y}^{ss}/G$, enabling cohomological computations.
• A modified form of Kirwan surjectivity is deduced: the Kirwan map $H^*_{G}(X) o H^*(X^{ss}/G)$ is surjective after passing to the Hochschild homology of the D-module categories, reflecting the geometry of the symplectic quotient.