# Albanese variety

In mathematics, the **Albanese variety** [math]A(V)[/math], named for Giacomo Albanese, is a generalization of the Jacobian variety of a curve.

## Contents

## Precise statement[edit]

The Albanese variety is the abelian variety [math]A[/math] generated by a variety [math]V[/math] taking a given point of [math]V[/math] to the identity of [math]A[/math]. In other words, there is a morphism from the variety [math]V[/math] to its Albanese variety [math]\operatorname{Alb}(V)[/math], such that any morphism from [math]V[/math] to an abelian variety (taking the given point to the identity) factors uniquely through [math]\operatorname{Alb}(V)[/math]. For complex manifolds, André Blanchard (1956) defined the Albanese variety in a similar way, as a morphism from [math]V[/math] to a torus [math]\operatorname{Alb}(V)[/math] 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[edit]

For compact Kähler manifolds the dimension of the Albanese variety is the Hodge number [math]h^{1,0}[/math], the dimension of the space of differentials of the first kind on [math]V[/math], which for surfaces is called the irregularity of a surface. In terms of differential forms, any holomorphic 1-form on [math]V[/math] is a pullback of translation-invariant 1-form on the Albanese variety, coming from the holomorphic cotangent space of [math]\operatorname{Alb}(V)[/math] at its identity element. Just as for the curve case, by choice of a base point on [math]V[/math] (from which to 'integrate'), an **Albanese morphism**

- [math] V \to \operatorname{Alb}(V) [/math]

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 [math]h^{1,0}[/math] and [math]h^{0,1}[/math] (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 [math]H^1(X, O_X).[/math] That [math]\dim \operatorname{Alb}(X) \leq h^{1,0}[/math] is a result of Jun-ichi Igusa in the bibliography.

## Roitman's theorem[edit]

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

- [math]CH_0(V) \to \operatorname{Alb}(V)(k)[/math]

from the Chow group of 0-dimensional cycles on *V* to the group of rational points of [math]\operatorname{Alb}(V)[/math], which is an abelian group since [math]\operatorname{Alb}(V)[/math] 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 [math]\operatorname{LAlb} (V)[/math] and have been proven by Luca Barbieri-Viale and Bruno Kahn (see the references III.13).

## Connection to Picard variety[edit]

The Albanese variety is dual to the Picard variety (the connected component of zero of the Picard scheme classifying invertible sheaves on *V*):

- [math]\operatorname{Alb} V = (\operatorname{Pic}_0 V)^\vee. [/math]

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

## See also[edit]

- Intermediate Jacobian
- Albanese scheme
- Motivic Albanese

## Notes & References[edit]

- ↑ Rojtman, A. A. (1980). "The torsion of the group of 0-cycles modulo rational equivalence".
*Annals of Mathematics*. Second Series**111**(3): 553–569. doi:10.2307/1971109. ISSN 0003-486X. - ↑ Bloch, Spencer (1979). "Torsion algebraic cycles and a theorem of Roitman".
*Compositio Mathematica***39**(1). http://www.numdam.org/item/CM_1979__39_1_107_0. - ↑ Spieß, Michael; Szamuely, Tamás (2003). "On the Albanese map for smooth quasi-projective varieties".
*Mathematische Annalen***325**: 1–17. doi:10.1007/s00208-002-0359-8. - ↑ Geisser, Thomas (2015). "Rojtman's theorem for normal schemes".
*Mathematical Research Letters***22**(4): 1129–1144. doi:10.4310/MRL.2015.v22.n4.a8.

- Barbieri-Viale, Luca; Kahn, Bruno (2016),
*On the derived category of 1-motives*, Astérisque,**381**, SMF, ISBN 978-2-85629-818-3, ISSN 0303-1179 - 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 - 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. doi:10.1073/pnas.41.5.317. PMID 16589672. Bibcode: 1955PNAS...41..317I. - Hazewinkel, Michiel, ed. (2001), "Albanese_variety",
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Albanese_variety

*https://en.wikipedia.org/wiki/Albanese variety was the original source. Read more*.