Mixed Hodge modules on stacks

TL;DR

This paper extends the theory of mixed Hodge modules to algebraic stacks using $$-categorical methods, demonstrating compatibility with six operations and weights, and relates motivic Hodge modules to geometric origin.

Contribution

It introduces a canonical extension of mixed Hodge modules to stacks and connects motivic Hodge modules with geometric origin within this framework.

Findings

Mixed Hodge modules extend to algebraic stacks with all six operations.

Drew's motivic Hodge modules embed fully faithfully into mixed Hodge modules.

Identification of the image as mixed Hodge modules of geometric origin.

Abstract

Using the $\infty$-categorical enhancement of mixed Hodge modules constructed by the author in a previous paper, we explain how mixed Hodge modules canonically extend to algebraic stacks, together with all the $6$ operations and weights. We also prove that Drew's approach to motivic Hodge modules gives an $\infty$-category that embeds fully faithfully in mixed Hodge modules, and we identify the image as mixed Hodge modules of geometric origin.