Albanese variety

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