Polynomial realizations of period matrices of projective smooth complete intersections and their deformation

TL;DR

This paper develops a polynomial-based interpretation of period integrals for smooth complete intersections, enabling explicit deformation formulas for period matrices using Bell polynomials and Maurer-Cartan equations.

Contribution

It introduces a novel polynomial framework for period integrals and derives explicit deformation formulas for period matrices of complete intersections.

Findings

Explicit formula for deformed period matrices in terms of original matrices and Bell polynomials.

Polynomial interpretation of period integrals as linear maps from polynomial rings.

Application of Maurer-Cartan equations in deformation theory of period integrals.

Abstract

Let $X$ be a smooth complete intersection over $\mathbb{C}$ of dimension $n-k$ in the projective space $\mathbf{P}^n_{\mathbb{C}}$, for given positive integers $n$ and $k$. For a given integral homology cycle $[\gamma] \in H_{n-k}(X(\mathbb{C}),\mathbb{Z})$, the period integral is defined to be a linear map from the de Rham cohomology group to $\mathbb{C}$ given by $[\omega] \mapsto \int_\gamma \omega$. The goal of this article is to interpret this period integral as a linear map from the polynomial ring with $n+k+1$ variables to $\mathbb{C}$ and use this interpretation to develop a deformation theory of period integrals of $X$. The period matrix is an invariant defined by the period integrals of the \textit{rational} de Rham cohomology, which compares the \textit{rational} structures ($\mathbb{Q}$-subspace structures) of the de Rham cohomology over $\mathbb{C}$ and the singular homology with coefficient $\mathbb{C}$. As a main result, when $X'$ is another projective smooth complete intersection variety deformed from $X$, we provide an explicit formula for the period matrix of $X'$ in terms of the period matrix of $X$ and the Bell polynomials evaluated at the deformation data. Our result can be thought of as a modern deformation theoretic treatment of the period integrals based on the Maurer-Cartan equation of a dgla (differential graded Lie algebra).