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 K3’s 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.