Abstract
Period mappings were introduced in the sixties [4] to study variation of complex structures of families of algebraic varieties. The theory of tautological systems was introduced recently [7,8] to understand period integrals of algebraic manifolds. In this paper, we give an explicit construction of a tautological system for each component of a period mapping. We also show that the D-module associated with the tautological system gives rise to many interesting vanishing conditions for period integrals at certain special points of the parameter space.
Highlights
Period mappings were introduced in the sixties [G] to study variation of complex structures of families of algebraic varieties
Let π : Y → B be the family of smooth CY hyperplane sections Ya ⊂ X, and let Htop be the Hodge bundle over B whose fiber at a ∈ B is the line Γ(Ya, ωYa) ⊂ Hd−1(Ya)
In [LY] the period integrals of this family are constructed by giving a canonical trivialization of Htop
Summary
We shall use τ to denote interchangeably both the D-module and its left defining ideal. The goal of this section is to write a system of scalar valued partial differential equations whose solution contains all the information of first order partial derivatives of period integrals. We get a system of differential equations whose solutions are vector valued of the form (φ1(a), . Equation (2.3f) shows that φ(a, b) is homogeneous of degree 1 in b, which implies that g = 0 and φ(a, b) = bkhk(a). Let M := DV ∨×V ∨/J where J is the left ideal generated by the operators in system (2.3). The Fourier transform of the D-module M = DV ∨×V ∨/J is M = DV ×V /J , where J is the DV ×V -ideal generated by the following operators:. The D-module M = DV ∨×V ∨/J is homogeneous since the ideal J is generated by homogeneous elements under the graduation deg deg. M is regular holonomic since its Fourier transform M is regular holonomic [Br]
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