Riemann form

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

• A lattice Λ in a complex vector space Cg.
• An alternating bilinear form α from Λ to the integers satisfying the following Riemann bilinear relations:

1. the real linear extension αR:Cg × Cg→R of α satisfies αR(iv, iw)=αR(v, w) for all (v, w) in Cg × Cg;
2. the associated hermitian form H(v, w)=αR(iv, w) + iαR(v, w) is positive-definite.

(The hermitian form written here is linear in the first variable.)

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 Cg/Λ admits the structure of an abelian variety if and only if there exists an alternating bilinear form α such that (Λ,α) is a Riemann form.

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