Period Mapping

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)

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

C^2

, namely, the line spanned by the vector

P

. The line

L

is determined by its slope, the ratio

:

au = rac{\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.

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	CATEGORIES ABOUT PERIOD MAPPING

	topological methods of algebraic geometry

	elliptic curves

Information About

Period Mapping