Albanese variety

In mathematics, the Albanese variety ${\displaystyle A(V)}$, named for Giacomo Albanese, is a generalization of the Jacobian variety of a curve.

Precise statement[]

The Albanese variety is the abelian variety ${\displaystyle A}$ generated by a variety ${\displaystyle V}$ taking a given point of ${\displaystyle V}$ to the identity of ${\displaystyle A}$. In other words, there is a morphism from the variety ${\displaystyle V}$ to its Albanese variety ${\displaystyle \operatorname {Alb} (V)}$, such that any morphism from ${\displaystyle V}$ to an abelian variety (taking the given point to the identity) factors uniquely through ${\displaystyle \operatorname {Alb} (V)}$. For complex manifolds, André Blanchard (1956) defined the Albanese variety in a similar way, as a morphism from ${\displaystyle V}$ to a torus ${\displaystyle \operatorname {Alb} (V)}$ 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 variety is the Hodge number ${\displaystyle h^{1,0}}$, the dimension of the space of differentials of the first kind on ${\displaystyle V}$, which for surfaces is called the irregularity of a surface. In terms of differential forms, any holomorphic 1-form on ${\displaystyle V}$ is a pullback of translation-invariant 1-form on the Albanese variety, coming from the holomorphic cotangent space of ${\displaystyle \operatorname {Alb} (V)}$ at its identity element. Just as for the curve case, by choice of a base point on ${\displaystyle V}$ (from which to 'integrate'), an Albanese morphism

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

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

Roitman's theorem[]

If the ground field k is algebraically closed, the Albanese map ${\displaystyle V\to \operatorname {Alb} (V)}$ can be shown to factor over a group homomorphism (also called the Albanese map)

${\displaystyle CH_{0}(V)\to \operatorname {Alb} (V)(k)}$

from the Chow group of 0-dimensional cycles on V to the group of rational points of ${\displaystyle \operatorname {Alb} (V)}$, which is an abelian group since ${\displaystyle \operatorname {Alb} (V)}$ is an abelian variety.

Roitman's theorem, introduced by A.A. Rojtman (1980), asserts that, for l prime to char(k), the Albanese map induces an isomorphism on the l-torsion subgroups.^[1][2] Replacing the Chow group by Suslin–Voevodsky algebraic singular homology after the introduction of Motivic cohomology Roitman's theorem has been obtained and reformulated in the motivic framework. For example, a similar result holds for non-singular quasi-projective varieties.^[3] Further versions of Roitman's theorem are available for normal schemes.^[4] Actually, the most general formulations of Roitman's theorem (i.e. homological, cohomological, and Borel–Moore) involve the motivic Albanese complex ${\displaystyle \operatorname {LAlb} (V)}$ and have been proven by Luca Barbieri-Viale and Bruno Kahn (see the references III.13).

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):

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

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

See also[]

• Intermediate Jacobian

Notes & References[]

• 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