, named for Giacomo Albanese, is a generalization of the Jacobian variety of a curve.
to an abelian variety (taking the given point to the identity) factors uniquely through
For complex manifolds, André Blanchard (1956) defined the Albanese variety in a similar way, as a morphism from
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.)
For compact Kähler manifolds the dimension of the Albanese variety is the Hodge number
is a pullback of translation-invariant 1-form on the Albanese variety, coming from the holomorphic cotangent space of
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.
If the ground field k is algebraically closed, the Albanese map
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.
[1][2] The constraint on the primality of the order of torsion to the characteristic of the base field has been removed by Milne[3] 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.
[4] Further versions of Roitman's theorem are available for normal schemes.
[5] 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).
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.