Albanese variety

In mathematics, the Albanese variety A(V), 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 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 A(V), such that any morphism from V to an abelian variety (taking the given point to the identity) factors uniquely through A(V). For complex manifolds Blanchard (1956) defined the Albanese variety in a similar way, as a morphism from V to a torus A(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.)

For compact Kähler manifolds the dimension of the Albanese 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, coming from the holomorphic cotangent space of 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

is defined, along which the 1-forms pull back. This morphism is unique up to a translation on the Albanese. 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 is dual to the Picard variety whose tangent space at the identity is given by ${\displaystyle H^{1}(X,O_{X})}$. That ${\displaystyle \dim Alb(X)\leq h^{1,0}}$ is a result of Igusa in the bibliography.

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