Hodge microsheaves on cotangent bundles and plumbings

TL;DR

The work constructs a Hodge-theoretic analogue of microsheaves for holomorphic exact symplectic manifolds, integrating mixed Hodge theory, Fourier transform techniques, and microlocal sheaf methods. It develops infinite-dimensional Hodge modules, saturation concepts, and a stable gluing framework to define Hodge microsheaves and their Wrapping theory, then connects these to Hodge structures on based loop spaces via Hain’s bar-construction approach. A core achievement is proving that endomorphism algebras of Hodge microsheaves on A_n-plumbings realize Koszul duality in a mixed-geometric sense, aligning with Etgü–Lekili’s results and yielding a categorical O interpretation from the Hodge-microlocal perspective. The constructions illuminate how mixed geometry underpins dualities between cocores and cores, offering a framework to transport Hodge structures to loop-space contexts and to extend Koszul duality to symplectic resolutions through explicit microlocal data and saturated categories.

Abstract

We introduce and study the category of Hodge microsheaves which is a Hodge-version of the category of microsheaves for a certain class of holomorphic exact symplectic manifolds. We then study Hodge-theoretic version of wrapped sheaves and discuss applications in topology and representation theory. Namely, we study (1) Hain's Hodge structures on the cohomology of based loop spaces of algebraic varieties, and (2) the Koszul duality of Ginzburg algebras by Etgü-Lekili from a mixed geometric perspective.

Theorems & Definitions (274)

• Theorem 1.1
• Conjecture 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 2.1
• Remark 2.2
• Definition 2.3
• Example 2.4: Tate Hodge structure
• Definition 2.5
• Proposition 2.6