de Rham Cohomology of Period Domains

TL;DR

This paper reviews the de Rham cohomology of period domains of Hodge structures, connecting ideas from differentiable stacks, toric geometry, and Hodge theory, with an emphasis on expository clarity and potential new relations.

Contribution

It provides an expository overview linking de Rham cohomology of period domains with differentiable stacks and toric compactifications, highlighting possible new conceptual connections.

Findings

Cohomology of period domains as differentiable stacks

Formulas for cohomology of toroidal compactifications

Connections between Hodge theory and toric geometry

Abstract

This is a review article discussing the de Rham cohomology of period domains of Hodge structures. We explain it as the de Rham cohomology of differentiable stacks as of a moduli space. We also discuss the cohomology of the partial toroidal compactification of these domains using known formulas on cohomology or Chow rings of toric structures. The text is expository and we have tried to connect some existing ideas that probably their relations not processed in the literature of Hodge theory. We state the significance of ideas as they naturally could be related, probably with not serious mathematical proof. The proofs stated in the text maybe expressed in a more serious context.