Period mapping

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In mathematics, in the field of algebraic geometry, the period mapping associates to a family of algebraic manifolds a family of Hodge structures.

The family of Hodge structures is given concretely by matrices of integrals. To illustrate these ideas, consider an elliptic curve $E$ with equation

$y^2 = x^3 + ax + b.\,$

Let $δ$ and $γ$ be an integral homology basis for $E$ where the intersection number $δ.γ = 1$ . Consider the differential 1-form

$\omega = \textrm{d}x/y.\,$

It is a holomorphic 1-form (differential of the first kind). Consider the integrals

$\int_\gamma \omega,\ \int_\delta \omega.\,$

These integrals are called periods. The vector of periods whose coordinates are the given integrals is the period matrix in this example. Denote the period matrix by $P$ . The matrix $P$ depends holomorphically on the parameters $a$ and $b$ of the elliptic curve. The map which sends $(a, b)$ to $P$ is a concrete representation the period map of the given family of elliptic curves.

Let us now make the connecton with Hodge structures. The holomorphic 1-form $ω$ defines a one-dimensional subspace of the complex cohomology of $E$ . Let us denote this subspace by $H 1,0$ . Thus we have a line $H 1,0$ in the two-dimensional complex vector space $H 1 (E, C)$ . The choice of homology basis $δ,γ$ defines an isomorphism of $H 1 (E, C)$ with the standard two-dimensional complex vector space $C 2$ . This isomorphism identifies $H 1,0$ with a line $L$ in $C 2$ , namely, the line spanned by the vector $P$ . The line $L$ is determined by its slope, the ratio