In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 C
g.
* An
alternating bilinear form
In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called ''scalars''). In other words, a bilinear form is a function that is linear i ...
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(''iv'', ''iw'')=α
R(''v'', ''w'') for all (''v'', ''w'') in C
g × C
g;
# the associated
hermitian form ''H''(''v'', ''w'')=α
R(''iv'', ''w'') + ''i''α
R(''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.
References
*
*
*
*
*
{{Bernhard Riemann
Abelian varieties
Bernhard Riemann