Albanese variety

An 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. 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.