# Albanese variety

In mathematics, the Albanese variety $A(V)$, named for Giacomo Albanese, is a generalization of the Jacobian variety of a curve.

## Precise statement

The Albanese variety of a smooth projective algebraic variety $V$ is an abelian variety $\operatorname {Alb} (V)$ together with a morphism $V\to \operatorname {Alb} (V)$ such that any morphism from $V$ to an abelian variety factors uniquely through this morphism. For complex manifolds, André Blanchard (1956) defined the Albanese variety in a similar way, as a morphism from $V$ to a complex torus $\operatorname {Alb} (V)$ such that any morphism to a complex torus factors uniquely through this map. (The complex torus $\operatorname {Alb} (V)$ need not be algebraic in this case.)

## Properties

For compact Kähler manifolds the dimension of the Albanese variety is the Hodge number $h^{1,0}$, the dimension of the space of differentials of the first kind on $V$, which for surfaces is called the irregularity of a surface. In terms of differential forms, any holomorphic 1-form on $V$ is a pullback of translation-invariant 1-form on the Albanese variety, coming from the holomorphic cotangent space of $\operatorname {Alb} (V)$ at its identity element. Just as for the curve case, by choice of a base point on $V$ (from which to 'integrate'), an Albanese morphism
    $V\to \operatorname {Alb} (V)$
is defined, along which the 1-forms pull back. This morphism is unique up to a translation on the Albanese variety. For varieties over fields of positive characteristic, the dimension of the Albanese variety may be less than the Hodge numbers $h^{1,0}$ and $h^{0,1}$ (which need not be equal). To see the former note that the Albanese variety is dual to the Picard variety, whose tangent space at the identity is given by $H^{1}(X,O_{X}).$ That $\dim \operatorname {Alb} (X)\leq h^{1,0}$ is a result of Jun-ichi Igusa in the bibliography.

## Roitman's theorem

If the ground field k is algebraically closed, the Albanese map $V\to \operatorname {Alb} (V)$ can be shown to factor over a group homomorphism (also called the Albanese map)
    $CH_{0}(V)\to \operatorname {Alb} (V)(k)$
from the Chow group of 0-dimensional cycles on V to the group of rational points of $\operatorname {Alb} (V)$, which is an abelian group since $\operatorname {Alb} (V)$ is an abelian variety.
Roitman's theorem, introduced by A.A. Rojtman (1980), asserts that, for l prime to char(k), the Albanese map induces an isomorphism on the l-torsion subgroups. The constraint on the primality of the order of torsion to the characteristic of the base field has been removed by Milne shortly thereafter: the torsion subgroup of $\operatorname {CH} _{0}(X)$ and the torsion subgroup of k-valued points of the Albanese variety of X coincide.
Replacing the Chow group by Suslin–Voevodsky algebraic singular homology after the introduction of Motivic cohomology Roitman's theorem has been obtained and reformulated in the motivic framework. For example, a similar result holds for non-singular quasi-projective varieties. Further versions of Roitman's theorem are available for normal schemes. Actually, the most general formulations of Roitman's theorem (i.e. homological, cohomological, and Borel–Moore) involve the motivic Albanese complex $\operatorname {LAlb} (V)$ and have been proven by Luca Barbieri-Viale and Bruno Kahn (see the references III.13).

## Connection to Picard variety

The Albanese variety is dual to the Picard variety (the connected component of zero of the Picard scheme classifying invertible sheaves on V):
    $\operatorname {Alb} V=(\operatorname {Pic} _{0}V)^{\vee }.$
For algebraic curves, the Abel–Jacobi theorem implies that the Albanese and Picard varieties are isomorphic.