Abstract

Abstract: This talk is about 1-parameter families of elliptic curves, K3 surfaces, and Calabi-Yau 3-folds—objects which arise, for example, in the theory of modular forms and in mirror symmetry—with particular attention to the role played by singular fibers. Instead of looking at the geometry of the family directly, one often studies the associated “variation of Hodge [i.e., complex analytic] structure”, and the degrees of related vector bundles on the parameter space are a tool for studying global behavior.

In his classic study of minimal elliptic fibrations, Kodaira described all possible singular fibers and their relation to the A-D-E classification (from Lie/singularity theory). We will first recall this and how one can relate fiber types to the Euler characteristic of the total space and the degree of the Hodge bundles.

What is interesting is how these relations generalize (or fail to generalize) to higher dimensions (K3, CY 3 -fold), and the related nonexistence (or existence) of non-isotrivial families with no singular fibers. We will describe some results along these (global) lines; if time permits, we will sketch how they fit with our earlier classification of (local) degenerations of CY threefold VHS ’s related to mirror symmetry, and lead to a Torelli theorem for the mirror quintic family.