Albanese variety

In mathematics, the Albanese variety A(V), named for Giacomo Albanese, is a generalization of the Jacobian variety of a curve, and is the abelian variety generated by a variety V. 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 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. (Although it is called a variety in this case, it need not be algebraic.)

For compact Kahler 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 an 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

$V\to \operatorname {Alb} (V)$

is defined, along which the 1-forms pull back. This morphism is unique up to a translation on the Albanese. 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).

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):

$\operatorname {Alb} \,V=(\operatorname {Pic} _{0}\,V)^{*}$

See also

• Intermediate Jacobian
• Albanese scheme

References

• Blanchard, André (1956), "Sur les variétés analytiques complexes", Annales Scientifiques de l'École Normale Supérieure. Troisième Série, 73: 157–202, ISSN 0012-9593, MR0087184
• P. Griffiths (1994). Principles of Algebraic Geometry. Wiley Classics Library. Wiley Interscience. pp. 331, 552. ISBN 0-471-05059-8. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
• Parshin, A. N. (2001) [1994], "Albanese variety", Encyclopedia of Mathematics, EMS Press