Speaker:	Bong Lian (Brandeis University)
Title:	Period Integrals and Tautological Systems
Date (JST):	Fri, Jun 08, 2012, 16:30 - 18:00
Place:	Room 002, Mathematical Sciences Building, Komaba Campus
Abstract:	We develop a global Poincar\'e residue formula to study period integrals of families of complex manifolds. For any compact complex manifold $X$ equipped with a linear system $V^*$ of generically smooth CY hypersurfaces, the formula expresses period integrals in terms of a canonical global meromorphic top form on $X$. Two important ingredients of this construction are the notion of a CY principal bundle, and a classification of such rank one bundles. We also generalize the construction to CY and general type complete intersections. When $X$ is an algebraic manifold having a sufficiently large automorphism group $G$ and $V^*$ is a linear representation of $G$, we construct a holonomic D-module that governs the period integrals. The construction is based in part on the theory of tautological systems we have developed earlier. The approach allows us to explicitly describe a Picard-Fuchs type system for complete intersection varieties of general types, as well as CY, in any Fano variety, and in a homogeneous space in particular. In addition, the approach provides a new perspective of old examples such as CY complete intersections in a toric variety or partial flag variety. The talk is based on recent joint work with R. Song and S.T. Yau.

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