The Borel-Serre Compactification for the Classifying Space of Hodge Structures

A 1-parameter variation of Hodge structures corresponds to a holomorphic, horizontal, locally liftable map into a classifying space of Hodge structures. In this paper it is shown that such a map has a limit in the reductive Borel-Serre compactification of the classifying space. The boundary component in which the limit lies is a union over possible polarizations of classifying spaces of Hodge structures on the primitive parts. It is discussed which boundary components can contain such limit points.