Generic Integral Manifolds for Weight Two Period Domains

Period integrals associated to an affine Delsarte type hypersurface

We calculate the period integrals for a special class of affine hypersurfaces (deformed Delsarte hypersurfaces) in an algebraic torus by the aid of their Mellin transforms. A description of the relation between poles of Mellin transforms of period integrals and the mixed Hodge structure of the cohomology of the hypersurface is given. By interpreting the period integrals as solutions to Pochhammer hypergeometric differential equation, we calculate concretely the irreducible monodromy group of period integrals that correspond to the compactification of the affine hypersurface in a complete simpl…

Computing period integrals of rigid double octic Calabi-Yau threefolds with Picard-Fuchs operator

We present a method for numerical computation of period integrals of a rigid Calabi-Yau threefold using Picard-Fuchs operator of a one-parameter smoothing. Our method gives a possibility of computing the lattice of period integrals of a rigid double octic without any explicit knowledge of its geometric properties, employing only simple facts from the theory of Fuchsian equations and computations in MAPLE with a library for differential equations. As a surprising consequence we also get approximations of additional integrals related to a singular (nodal) model of considered Calabi-Yau threefold…

Period integrals and Hodge modules

We define a map $\mathcal{P}_M$ attached to any polarized Hodge module $M$ such that the restriction of $\mathcal{P}_M$ to a locus on which $M$ is a variation of Hodge structures induces the usual period integral pairing for this variation of Hodge structures. In the case that $M$ is the minimal extension of a simple polarized variation of Hodge structures $V$, we show that the homotopy image of $\mathcal{P}_M$ is the minimal extension of the graph morphism of the usual period integral map for $V$.

Determinant of Period Integral

Period integrals arise from the comparison between Betti cohomologies and de Rham cohomologies. In this paper, we give a formula for the determinant of period integrals in terms of the canonical pairing between Picard group of rank 1 realizations and relative Chow group. It is compatible with the similar formula for Galois modules proved by the first author, and as a consequence, under the assumption of Gersten resolution, the determinant of the cohomology of a determinantally motivic object is motivic.

关于我对周期积分研究的一些评论

Some comments on my study of period-integrals

本文以无证明形式介绍作者近期利用(a,b)-模研究周期积分渐近展开奇异项的结果，并阐述该代数结构的有趣性和实用性。

Hasse-Witt matrices, unit roots and period integrals

Motivated by the work of Candelas, de la Ossa and Rodriguez-Villegas [6], we study the relations between Hasse-Witt matrices and period integrals of Calabi-Yau hypersurfaces in both toric varieties and partial flag varieties. We prove a conjecture by Vlasenko [23] on higher Hasse-Witt matrices for toric hypersurfaces following Katz's method of local expansion [14, 15]. The higher Hasse-Witt matrices also have close relation with period integrals. The proof gives a way to pass from Katz's congruence relations in terms of expansion coefficients [15] to Dwork's congruence relations [8] about peri…