Pure Maths Colloquium: Local and Global Hodge theory of Calabi-Yau fibrations
28 January 2008 16:15 in CM221
This talk is about 1-parameter families of elliptic curves, K3 surfaces, and CY 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 structure (VHS). This is a linear-algebraic object which keeps track of how period integrals change as the algebraic variety deforms. 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 (1960's), Kodaira described all possible singular fibers and their relation to the A-D-E classification (from Lie/singularity theory). I want to spend quite some time explaining 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, and also briefly explain our own classification of (local) degenerations of CY 3-fold VHS's related to mirror symmetry. This is joint work with P. Griffiths and M. Green.
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