Albanese variety
In mathematics, the Albanese variety formula_1, named for Giacomo Albanese, is a generalization of the Jacobian variety of a curve. Precise statement. The Albanese variety is the abelian variety formula_2 generated by a variety formula_3 taking a given point of formula_3 to the identity of formula_2. In other words, there is a morphism from the variety formula_3 to its Albanese variety formula_7, such that any morphism from formula_3 to an abelian variety (taking the given point to the identity) factors uniquely through formula_7. For complex manifolds, defined the Albanese variety in a similar way, as a morphism from  formula_3 to a torus formula_7 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 formula_12, the dimension of the space of differentials of the first kind on formula_3, which for surfaces is called the irregularity of a surface. In terms of differential forms, any holomorphic 1-form on formula_3 is a pullback of translation-invariant 1-form on the Albanese variety, coming from the holomorphic cotangent space of formula_7  at its identity element. Just as for the curve case, by choice of a base point on formula_3 (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 formula_12 and formula_19 (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 formula_20 That  formula_21 is a result of Jun-ichi Igusa in the bibliography. Roitman's theorem. If the ground field "k" is algebraically closed, the Albanese map formula_17 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 formula_7, which is an abelian group since formula_7 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. 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 formula_26 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.