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

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).