Extended period domains, algebraic groups, and higher Albanese manifolds
  For a linear algebraic group G over the field of rational numbers, we consider the period domains D classifying G-mixed Hodge structures, and construct the extended period domains as toroidal partial compactifications. We give an interpretation of higher Albanese manifolds by Hain and Zucker by using the above D for some G, and extend them via the toroidal partial compactifications. 
Kazuya Kato, Chikara Nakayama, Sampei Usui
math.AG