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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 positiveintegers $n$ and $k$. For a given integral homology cycle $[\gamma] \inH_{n-k}(X(\mathbb{C}),\mathbb{Z})$, the period integral is defined to be alinear 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 thisperiod integral as a linear map from the polynomial ring with $n+k+1$ variablesto $\mathbb{C}$ and use this interpretation to develop a deformation theory ofperiod integrals of $X$. The period matrix is an invariant defined by theperiod integrals of the \textit{rational} de Rham cohomology, which comparesthe \textit{rational} structures ($\mathbb{Q}$-subspace structures) of the deRham cohomology over $\mathbb{C}$ and the singular homology with coefficient$\mathbb{C}$. As a main result, when $X'$ is another projective smooth completeintersection variety deformed from $X$, we provide an explicit formula for theperiod matrix of $X'$ in terms of the period matrix of $X$ and the Bellpolynomials evaluated at the deformation data. Our result can be thought of asa modern deformation theoretic treatment of the period integrals based on theMaurer-Cartan equation of a dgla (differential graded Lie algebra).

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