Period Integrals of CY and General Type Complete Intersections
Bong H. Lian, Shing-Tung Yau

TL;DR
This paper develops a global residue formula to analyze period integrals of CY and general type complete intersections, providing explicit systems like Picard-Fuchs equations in various geometric contexts.
Contribution
It introduces a new global Poincaré residue formula for period integrals of CY and general type complete intersections, extending previous methods and incorporating automorphism groups and tautological systems.
Findings
Explicit Picard-Fuchs systems for complete intersections in Fano varieties
Construction of a holonomic D-module governing period integrals
Application to CY and general type varieties in homogeneous spaces
Abstract
We develop a global Poincar\'e residue formula to study period integrals of families of complex manifolds. For any compact complex manifold equipped with a linear system of generically smooth CY hypersurfaces, the formula expresses period integrals in terms of a canonical global meromorphic top form on . 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 is an algebraic manifold having a sufficiently large automorphism group and is a linear representation of , 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…
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