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Period integrals of CY and general type complete intersections

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Abstract

We develop a global Poincaré 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 in the paper Lian, Song and Yau (arXiv:1105.2984v1, 2011). 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.

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Authors and Affiliations

  1. Department of Mathematics, Brandeis University, Waltham, MA, 02454, USA

    Bong H. Lian

  2. Department of Mathematics, Harvard University, Cambridge, MA, 02138, USA

    Shing-Tung Yau

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  1. Bong H. Lian
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  2. Shing-Tung Yau
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Correspondence to Bong H. Lian.

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Lian, B.H., Yau, ST. Period integrals of CY and general type complete intersections. Invent. math. 191, 35–89 (2013). https://doi.org/10.1007/s00222-012-0391-6

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  • DOI: https://doi.org/10.1007/s00222-012-0391-6

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