Riemann form

In mathematics, a Riemann form in the theory of abelian varieties and modular forms, is the following data:

A Lattice #Lattices in [complex space|lattice] Λ in a complex vector space C^g.
An alternating bilinear form α from Λ to the integers satisfying the following Riemann bilinear relations:

the real linear extension α_R:C^g × C^g→R of α satisfies α_R=α_R for all in C^g × C^g;
the associated hermitian form H=α_R + iα_R is positive-definite.

Riemann forms are important because of the following:

The alternatization of the Chern class of any factor of automorphy is a Riemann form.
Conversely, given any Riemann form, we can construct a factor of automorphy such that the alternatization of its Chern class is the given Riemann form.

Furthermore, the complex torus C^g/Λ admits the structure of an abelian variety if and only if there exists an alternating bilinear form α such that is a Riemann form.