Hodge microsheaves on cotangent bundles and plumbings

TL;DR

This paper develops the theory of Hodge microsheaves on cotangent bundles, connecting symplectic geometry with Hodge theory, and explores their applications in topology and representation theory.

Contribution

It introduces Hodge microsheaves as a Hodge-variant of microsheaves for holomorphic symplectic manifolds, and studies their implications in Hodge structures and Koszul duality.

Findings

Hain's Hodge structures on loop space cohomology analyzed

Koszul duality of Ginzburg algebras examined from a geometric perspective

Applications in topology and representation theory discussed

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 Etg\"u-Lekili from a mixed geometric perspective.