Mixed Hodge modules on stacks
Swann Tubach
TL;DR
The paper addresses extending mixed Hodge modules to algebraic stacks while preserving the six functor formalism, duality, and weights, and it connects this Hodge-theoretic framework with motivic constructions. It leverages an infinity-category enhancement of mixed Hodge modules to build a robust D_H(-) that glues along stacks and descends from schemes, and it couples this with Drew's motivic Hodge modules, showing a faithful embedding of motivic Hodge modules into mixed Hodge modules of geometric origin. A key contribution is the demonstration that motivic Hodge modules can be realized as modules over an algebra object in étale motives, with a fully faithful inclusion into the Hodge world and an equivalence on geometric-origin subcategories. The work also develops nearby and vanishing cycles for stacks, a weight theory in the stack context (with caveats for general stacks), and a detailed comparison to existing equivariant and motivic constructions, providing a comprehensive toolbox for studying Hodge cohomology of stacks.
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.