 
Summary: Cycles, Motives and Shimura Varieties
Editor: V. Srinivas
Copyright c 2010 Tata Institute of Fundamental Research
Publisher: Narosa Publishing House, New Delhi, India
Mixed Hodge Structures Associated to
Geometric Variations
Donu Arapura
This is largely an exposition of Morihiko Saito's theory of mixed Hodge
modules, which gives a very general framework in which to do Hodge theory.
A mixed Hodge module over a point is the same thing as a mixed Hodge
structure, and in general the category of these objects possess operations
which mirror the standard operations of sheaf theory. Saito's theory has
been laid out in a pair of long and densely written papers [S1, S3], and it
would be fair to say that the details have not been very widely assimilated.
The primary goal of this article is to make these ideas more accessible and
explicit.
A variation of Hodge structure gives a good notion of a regular family of
Hodge structures. It is natural to extend this to families with singularities.
Recall that the constituents of a variation of a Hodge structure are a local
system and a compatible filtered vector bundle with connection. As a first
