Inmathematics, theAlbanese variety${\displaystyle A(V)}$, named forGiacomo Albanese, is a generalization of theJacobian variety of a curve.

The Albanese variety of a smooth projective algebraic variety${\displaystyle V}$ is an abelian variety${\displaystyle \mathrm{Alb}(V)}$ together with a morphism${\displaystyle V\to \mathrm{Alb}(V)}$ such that any morphism from${\displaystyle V}$ to an abelian variety factors uniquely through this morphism. For complex manifolds, André Blanchard (1956) defined the Albanese variety in a similar way, as a morphism from${\displaystyle V}$ to a complex torus${\displaystyle \mathrm{Alb}(V)}$ such that any morphism to a complex torus factors uniquely through this map. (The complex torus${\displaystyle \mathrm{Alb}(V)}$ need not be algebraic in this case.)

ForcompactKähler manifolds the dimension of the Albanese variety is theHodge number${\displaystyle h^{1,0}}$, the dimension of the space ofdifferentials of the first kind on${\displaystyle V}$, which for surfaces is called theirregularity of a surface. In terms ofdifferential forms, any holomorphic 1-form on${\displaystyle V}$ is apullback of translation-invariant 1-form on the Albanese variety, coming from the holomorphiccotangent space of${\displaystyle \mathrm{Alb}(V)}$ at its identity element. Just as for the curve case, by choice of abase point on${\displaystyle V}$ (from which to 'integrate'), anAlbanese morphism

${\displaystyle V\to \mathrm{Alb}(V)}$

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${\displaystyle h^{1,0}}$ and${\displaystyle h^{0,1}}$ (which need not be equal). To see the former note that the Albanese variety is dual to thePicard variety, whose tangent space at the identity is given by${\displaystyle H^1(X,O_X).}$ That${\displaystyle \dim \mathrm{Alb}(X)\le h^{1,0}}$ is a result ofJun-ichi Igusa in the bibliography.

If the ground fieldk isalgebraically closed, the Albanese map${\displaystyle V\to \mathrm{Alb}(V)}$ can be shown to factor over a group homomorphism (also called theAlbanese map)

${\displaystyle C{H}_0(V)\to \mathrm{Alb}(V)(k)}$

from theChow group of 0-dimensional cycles onV to the group ofrational points of${\displaystyle \mathrm{Alb}(V)}$, which is an abelian group since${\displaystyle \mathrm{Alb}(V)}$ is an abelian variety.

Roitman's theorem, introduced by A.A. Rojtman (1980), asserts that, forl prime to char(k), the Albanese map induces an isomorphism on thel-torsion subgroups.^[1][2]The constraint on the primality of the order of torsion to the characteristic of the base field has been removed by Milne^[3] shortly thereafter: the torsion subgroup of${\displaystyle {\mathrm{CH}}_0(X)}$ and the torsion subgroup ofk-valued points of the Albanese variety ofX coincide.

Replacing the Chow group by Suslin–Voevodsky algebraic singular homology after the introduction ofMotivic cohomologyRoitman's theorem has been obtained and reformulated in the motivic framework. For example, a similar result holds for non-singular quasi-projective varieties.^[4] Further versions ofRoitman's theorem are available for normal schemes.^[5] Actually, the most general formulations ofRoitman's theorem (i.e. homological, cohomological, andBorel–Moore) involve the motivic Albanese complex${\displaystyle \mathrm{LAlb}(V)}$ and have been proven by Luca Barbieri-Viale and Bruno Kahn (see the references III.13).

The Albanese variety isdual to thePicard variety (theconnected component of zero of thePicard scheme classifyinginvertible sheaves onV):

${\displaystyle \mathrm{Alb}V=({\mathrm{Pic}}_{0}V{)}^{\vee }.}$

For algebraic curves, theAbel–Jacobi theorem implies that the Albanese and Picard varieties are isomorphic.

• Intermediate Jacobian
• Albanese scheme
• Motivic Albanese

• Barbieri-Viale, Luca; Kahn, Bruno (2016),On the derived category of 1-motives, Astérisque, vol. 381, SMF,arXiv:1009.1900,ISBN 978-2-85629-818-3,ISSN 0303-1179,MR 3545132
• Blanchard, André (1956), "Sur les variétés analytiques complexes",Annales Scientifiques de l'École Normale Supérieure, Série 3,73 (2):157–202,doi:10.24033/asens.1045,ISSN 0012-9593,MR 0087184
• Griffiths, Phillip;Harris, Joe (1994).Principles of Algebraic Geometry. Wiley Classics Library. Wiley Interscience. pp. 331, 552.ISBN 978-0-471-05059-9.
• Igusa, Jun-ichi (1955)."A fundamental inequality in the theory of Picard varieties".Proceedings of the National Academy of Sciences of the United States of America.41 (5):317–20.Bibcode:1955PNAS...41..317I.doi:10.1073/pnas.41.5.317.PMC 528086.PMID 16589672.
• Parshin, Aleksei N. (2001) [1994],"Albanese_variety",Encyclopedia of Mathematics,EMS Press

• Abelian varieties

• Articles with short description
• Short description matches Wikidata