Period mapping

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 $delta$ and $gamma$ be an integral homology basis for $E$ where the intersection number $delta\; .\; gamma\; =\; 1$. Consider the differential 1-form

:$omega\; =\; extrm\{d\}x/y.,$

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 $omega$ 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 $delta,\; gamma$ 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

:$au\; =\; frac\{int\_gamma\; omega\}\{int\_delta\; omega\}.,$

The period map in this context is the map

:$(a,b)\; o\; au$

modulo the choice of homology basis subject to the constraint on the intersection number.

External links

* [http://eom.springer.de/P/p072140.htm]

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