Travaux de Griffiths

By the work of Griffiths, the period map of the cohomology
of a family of smooth complex projective varieties is holomorphic and
satisfies a first order differential condition called Griffiths'
transversality. Many of the deepest results in Hodge theory known
to date depend critically upon the fact that we have a complete
description of the asymptotic behavior of such a period map near
the boundary which depends only on the local monodromy. This in
turn is a consequence of the fact that the holomorphic sectional
curvature along a period map is negative and bounded away from
zero. In this talk, I will discuss classifying spaces of Hodge
structure and outline the computation of the curvature of the
classifying space following Deligne's Travaux de Griffiths.