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 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.)

Properties

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

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). To see the former note that the Albanese is dual to the Picard variety whose tangent space at the identity is given by $H^1(X, O_X)$ . That $\dim X \leq h^{1,0}$ is a result of Igusa in the bibliography.

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.

References

Blanchard, André (1956), "Sur les variétés analytiques complexes", Annales Scientifiques de l'École Normale Supérieure. Troisième Série 73: 157–202, ISSN 0012-9593, MR 0087184
P. Griffiths; J. Harris (1994), Principles of Algebraic Geometry, Wiley Classics Library, Wiley Interscience, pp. 331, 552, ISBN 0-471-05059-8
Parshin, A. N. (2001), "Albanese_variety", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Igusa, Jun-ichi (1955), A fundamental inequality in the theory of Picard varieties

Albanese variety

Precise statement

Properties

Connection to Picard variety

See also

References