View source for Albanese variety

An
[[Abelian variety|Abelian variety]] ${\rm Alb}(X)$ canonically attached to an
algebraic variety $X$, which is the solution of the following
universal problem: There exists a morphism $\phi:X\to{\rm Alb}(X)$ such that any morphism
$f:X\to A$ into an Abelian variety $A$ factors into a product $f={\tilde f}\phi$, where ${\tilde f}:A\to{\rm Alb}(X)$
(so named in honour of G. Albanese). If $X$ is a complete non-singular
variety over the field of complex numbers, the Albanese variety can be
described as follows. Let $\Omega^1$ be the space of everywhere-regular
differential forms of degree 1 on $X$. Each one-dimensional cycle $\gamma$
of the topological space $X$ determines a linear function $\omega\mapsto \int_\gamma\omega$ on
$\Omega^1$. The image of the mapping $H_1(X,{\mathbb Z}) \to (\Omega^1)^*$ thus obtained is a lattice $\Gamma$ in
$(\Omega^1)^*$, and the quotient space $(\Omega^1)^*/\Gamma$ coincides with the Albanese variety of
$X$. From the algebraic point of view, an Albanese variety may be
considered as a method of defining an algebraic structure on some
quotient group of the group ${\mathbb Z}$ of zero-dimensional cycles of degree 0
on $X$. If $X$ is a non-singular complete algebraic curve, both its
Picard variety and its Albanese variety are called its
[[Jacobi variety|Jacobi variety]]. If the ground field has
characteristic zero, then the equalities 
$${\rm dim}\;{\rm Alb}(X) = {\rm dim}_k\; H^0(X,\Omega_X^1) = {\rm dim}_k\; H^1(X,{\mathcal O}_X) $$
are valid. The number
${\rm dim}\;{\rm Alb}(X)$ is called the irregularity
${\rm irr}(X)$ of the variety $X$. If the field has
finite characteristic, the inequalities 
$${\rm irr}\;(X) \le {\rm dim}\; H^0(X,\Omega_X^1) \text{ and } {\rm irr}\;(X)\le {\rm dim}\; H^1(X,{\mathcal O}_X) $$
hold. If the ground
field has positive characteristics it can happen that 
$${\rm dim}\; H^0(X,\Omega_X^1) \ne  {\rm dim}\; H^1(X,{\mathcal O}_X) $$
The
Albanese variety is dual to the
[[Picard variety|Picard variety]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD
valign="top"> M. Baldassarri, "Algebraic varieties" , Springer
(1956)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">
S. Lang, "Abelian varieties" , Springer (1983)</TD></TR></table>