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Period Integrals of CY and General Type Complete Intersections

May, 2011
60 pages
Published in:
  • Invent.Math. 191 (2013) 1
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Abstract: (arXiv)
We develop a global Poincar\'e residue formula to study period integrals of families of complex manifolds. For any compact complex manifold XX equipped with a linear system V∗V^* of generically smooth CY hypersurfaces, the formula expresses period integrals in terms of a canonical global meromorphic top form on XX. Two important ingredients of our construction are the notion of a CY principal bundle, and a classification of such rank one bundles. We also generalize our construction to CY and general type complete intersections. When XX is an algebraic manifold having a sufficiently large automorphism group GG and V∗V^* is a linear representation of GG, 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 in the paper \cite{LSY1}, joint with R. Song. 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.
Note:
  • An erratum is included to correct Theorem 3.12 (Uniqueness of CY structure)
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