Extended period domains and algebraic groups

TL;DR

This paper introduces extended period domains for linear algebraic groups, providing compactifications of various mixed Hodge structure moduli spaces and related geometric objects.

Contribution

It constructs the extended period domains $D_{oldsymbol{ au}}$ and offers toroidal compactifications for mixed Mumford--Tate domains and Shimura varieties.

Findings

Construction of extended period domains $D_{oldsymbol{ au}}$

Toroidal partial compactifications of mixed Shimura varieties

Application to higher Albanese manifolds

Abstract

For a linear algebraic group $G$ over $\mathbf Q$, we consider the period domains $D$ classifying $G$-mixed Hodge structures, and construct the extended period domains $D_{\Sigma}$. In particular, we give toroidal partial compactifications of mixed Mumford--Tate domains, mixed Shimura varieties over $\mathbf C$, and higher Albanese manifolds.