Albanese variety

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

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]

  1. 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. 
  2. 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. 
  3. 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. 
  4. Geisser, Thomas (2015). "Rojtman's theorem for normal schemes". Mathematical Research Letters 22 (4): 1129–1144. doi:10.4310/MRL.2015.v22.n4.a8. 

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