Moduli and Hodge Theory

Phillip Griffiths
Institute for Advanced Studies, Princeton (USA)
Moduli spaces of varieties
are of central interest in algebraic geometry. For
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.