Towards a L 2 cohomology theory for Hodge modules on infinite covering spaces: L 2 constructible cohomology and L 2 de Rham cohomology for coherent D-modules

TL;DR

This paper develops a framework for L2 cohomology theories for Hodge modules on infinite covering spaces, extending previous work on L2-cohomology and proposing conjectures related to L2-Mixed Hodge structures.

Contribution

It introduces Von Neumann invariants for constructible complexes and coherent D-modules, and formulates conjectures linking L2 cohomology with Saito's Mixed Hodge Modules.

Findings

Constructed Von Neumann invariants for complexes and D-modules.

Formulated conjectural generalization of L2-Mixed Hodge structures.

Provided partial results supporting the conjectures.

Abstract

This article constructs Von Neumann invariants for constructible complexes and coherent D-modules on compact complex manifolds, generalizing the work of the author on coherent L 2-cohomology. We formulate a conjectural generalization of Dingoyan's L 2-Mixed Hodge structures in terms of Saito's Mixed Hodge Modules and give partial results in this direction. 2020 AMS Classification: 32J27.