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
- y2 = x3 + ax + b.
Let δ and γ be an integer homology basis for E where the intersection number δ.γ = 1. Consider the differential 1-form
- ω = dx / y.
It is a holomorphic 1-form (differential of the first kind). Consider the integrals
.
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 H1,0. Thus we have a line H1,0 in the two-dimensional complex vector space H1(E,C). The choice of homology basis δ,γ defines an isomorphism of H1(E,C) with the standard two-dimensional complex vector space C2. This isomorphism identifies H1,0 with a line L in C2, namely, the line spanned by the vector P. The line L is determined by its slope, the ratio
-
τ = ∫ ω / ∫ ω. γ δ
The period map in this context is the map
- (a,b) − > τ
modulo the choice of homology basis subject to the constraint on the intersection number.
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