Moduli and Hodge Theory

Phillip Griffiths
Institute for Advanced Studies, Princeton (USA)
Abstract:
Moduli spaces of varieties
\[X\]
are of central interest in algebraic geometry. For
\[X\]
smooth and of general type the moduli space
\[\it M\]
exists and has a canonical projective completion
\[\overline{\it M}\]
. Aside from the case of algebraic curves there are essentially no general results about or examples of the structure of the boundary
\[\overline{\it M}\setminus {\it M}\]
.
Hodge theory provides the basic invariant of a complex algebraic variety. Using Lie theory the space of Hodge structures and its boundary is well understood. It is therefore natural to use the Hodge-theoretic boundary to study
\[\overline{\it M}\setminus {\it M}\]
. This talk will give an informal presentation of this approach together with one result and one application to moduli of a particularly interesting algebraic surface. *Based on joint work in progress with Mark Green, Radu Laza, and Colleen Robles.