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 is the abelian variety $$A$$ generated by a variety $$V$$ taking a given point of $$V$$ to the identity of $$A$$. In other words, there is a morphism from the variety $$V$$ to its Albanese variety $$\operatorname{Alb}(V)$$, such that any morphism from $$V$$ to an abelian variety (taking the given point to the identity) factors uniquely through $$\operatorname{Alb}(V)$$. For complex manifolds, defined the Albanese variety in a similar way, as a morphism from $$V$$ to a torus $$\operatorname{Alb}(V)$$ 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 $$h^{1,0}$$, 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, 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.