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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