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
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
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