Hodge-to-de Rham degeneration for stacks

TL;DR

This paper extends the Hodge-to-de Rham degeneration theorem to a new class of stacks by introducing Hodge-proper stacks, generalizing spreading techniques, and applying these to equivariant cohomology.

Contribution

It introduces Hodge-proper stacks and generalizes spreading out results to higher Artin stacks, enabling the extension of Hodge-to-de Rham degeneration to these objects.

Findings

Hodge-to-de Rham degeneration holds for Hodge-proper stacks.

Proper and some global quotient stacks are Hodge-properly spreadable.

A non-canonical Hodge decomposition for equivariant cohomology is established.

Abstract

We introduce a notion of a Hodge-proper stack and extend the method of Deligne-Illusie to prove the Hodge-to-de Rham degeneration in this setting. In order to reduce the statement in characteristic $0$ to characteristic $p$, we need to find a good integral model of a stack (a so-called spreading), which, unlike in the case of schemes, need not to exist in general. To address this problem we investigate the property of spreadability in more detail by generalizing standard spreading out results for schemes to higher Artin stacks and showing that all proper and some global quotient stacks are Hodge-properly spreadable. As a corollary we deduce a (non-canonical) Hodge decomposition of the equivariant cohomology for certain classes of varieties with an algebraic group action.