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.)
For compact Kähler manifolds the dimension of the Albanese is the Hodge number h1,0, 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
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 h1,0 and h0,1 (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 . That is a result of Igusa in the bibliography.
If the ground field k is algebraically closed, the Albanese map can be shown to factor over a group homomorphism (also called the Albanese map)
Roitman's theorem, introduced by Rojtman (1980), asserts that, for l prime to char(k), the Albanese map induces an isomorphism on the l-torsion subgroups. 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. Further versions of Roitman's theorem are available for normal schemes. Actually, the most general formulations of Roitman's theorem (i.e. homological, cohomological & Borel-Moore) involve the Motivic Albanese complex LAlb (V) and have been proven by L. Barbieri-Viale and B. Kahn (see the references III.13).
Connection to Picard variety
For algebraic curves, the Abel–Jacobi theorem implies that the Albanese and Picard varieties are isomorphic.
Notes & References
- 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, MR 577137
- S. Bloch. Torsion algebraic cycles and a theorem of Roitman. Compositio Mathematica Vol. 39, Fasc. 1, 1979
- M. Spieß & T. Szamuely On the Albanese map for smooth quasi-projective varieties Math. Ann. 325 (2003), no. 1, 1–17
- T. Geisser Rojtman’s theorem for normal schemes Mathematical Research Letters Vol. 22 N. 4 (2015) 1129 – 1144
- L. Barbieri-Viale; B. Kahn (2016), On the derived category of 1-motives, Astérisque, 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. 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
- Igusa, Jun-ichi (1955), "A fundamental inequality in the theory of Picard varieties", Proc. Natl. Acad. Sci. U.S.A., 41: 317–20, PMC , PMID 16589672
- Parshin, A. N. (2001), "Albanese_variety", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4