In mathematics, the Albanese variety , named for Giacomo Albanese, is a generalization of the Jacobian variety of a curve.

Precise statement
The Albanese variety is the abelian variety  generated by a variety  taking a given point of  to the identity of . In other words, there is a morphism from the variety  to its Albanese variety , such that any morphism from  to an abelian variety (taking the given point to the identity) factors uniquely through . For complex manifolds,  defined the Albanese variety in a similar way, as a morphism from  to a torus  such that any morphism to a torus factors uniquely through this map. (It is an analytic variety in this case; it need not be algebraic.)

Properties
For compact Kähler manifolds the dimension of the Albanese variety is the Hodge number , the dimension of the space of differentials of the first kind on , which for surfaces is called the irregularity of a surface. In terms of differential forms, any holomorphic 1-form on  is a pullback of translation-invariant 1-form on the Albanese variety, coming from the holomorphic cotangent space of  at its identity element. Just as for the curve case, by choice of a base point on  (from which to 'integrate'), an Albanese morphism

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  and  (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  That  is a result of Jun-ichi Igusa in the bibliography.

Roitman's theorem
If the ground field k is algebraically closed, the Albanese map  can be shown to factor over a group homomorphism (also called the Albanese map)

from the Chow group of 0-dimensional cycles on V to the group of rational points of , which is an abelian group since  is an abelian variety.

Roitman's theorem, introduced by , 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   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  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):

For algebraic curves, the Abel–Jacobi theorem implies that the Albanese and Picard varieties are isomorphic.

See also
Intermediate Jacobian
Albanese scheme
Motivic Albanese

Notes & references

Abelian varieties