Hodge Theory and Moduli Theory1
Phillip Griffiths (Institute of Advanced Studies)
Hodge theory and
moduli theory are two of the most active and important areas in algebraic geometry.
For the cases of curves, abelian varieties and K3s the relation between them
is classical and much studied; for Calabi-Yau threefolds there is also work inspired by mirror
symmetry. In contrast, for surfaces of general type the moduli space M has
been constructed by Kollār, Shepherd-Barron, and Alexeev, but essentially no examples seem to be known
describing its global structure, especially the configuration of boundary
components of M. By way of
contrast, using methods from representation theory the study of the moduli
space Γ\D for equivalence
classes of polarized Hodge structures has recently seen considerable
developments. In these talks we will discuss two examples of the emerging and
in some cases tight connection between moduli theory and Hodge theory, where
the latter serves as a guide to the former. The topics of the lectures will
be algebraic surfaces X with pg(X) = 2 and small KX2 , an extension of the Satake-Baily-Borel
compactification to the first non-classical case Γ\D* of polarized Hodge structures of weight 2 and with h2,0 = 2, and combining the
two using the extended period map M
→ Γ\D* to illustrate how
Hodge theory may be used to help understand the KSBA moduli spaces. The talks
will be built around the special examples mentioned above.
1 Much of the material in the lectures
will be based on joint work with Mark Green, Radu Laza and Colleen Robles.