Hodge-to-de Rham degeneration for stacks

Kubrak, Dmitry; Prikhodko, Artem

doi:10.1093/imrn/rnab054

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Hodge-to-de Rham degeneration for stacks

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Kubrak, Dmitry
Max Planck Institute for Mathematics, Max Planck Society;

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https://doi.org/10.1093/imrn/rnab054
(Publisher version)

https://doi.org/10.48550/arXiv.1910.12665
(Preprint)

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Citation

Kubrak, D., & Prikhodko, A. (2022). Hodge-to-de Rham degeneration for stacks. International Mathematics Research Notices, 2022(17), 12852-12939. doi:10.1093/imrn/rnab054.

Cite as: https://hdl.handle.net/21.11116/0000-000A-1579-7

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.