Albanese variety

Last updated

In mathematics, the Albanese variety, named for Giacomo Albanese, is a generalization of the Jacobian variety of a curve.

Contents

Precise statement

The Albanese variety is the abelian variety generated by a variety taking a given point of to the identity of . In other words, there is a morphism from the variety to its Albanese variety , such that any morphism from to an abelian variety (taking the given point to the identity) factors uniquely through . For complex manifolds, AndréBlanchard ( 1956 ) defined the Albanese variety in a similar way, as a morphism from to a torus 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 , the dimension of the space of differentials of the first kind on , which for surfaces is called the irregularity of a surface. In terms of differential forms, any holomorphic 1-form on is a pullback of translation-invariant 1-form on the Albanese variety, coming from the holomorphic cotangent space of at its identity element. Just as for the curve case, by choice of a base point on (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 variety. For varieties over fields of positive characteristic, the dimension of the Albanese variety may be less than the Hodge numbers and (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 That is a result of Jun-ichi Igusa in the bibliography.

Roitman's theorem

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)

from the Chow group of 0-dimensional cycles on V to the group of rational points of , which is an abelian group since 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] 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 and the torsion subgroup of k-valued points of the Albanese variety of X coincide.

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. [4] Further versions of Roitman's theorem are available for normal schemes. [5] Actually, the most general formulations of Roitman's theorem (i.e. homological, cohomological, and Borel–Moore) involve the motivic Albanese complex 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):

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

See also

Notes & references

  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. JSTOR   1971109. MR   0577137.
  2. Bloch, Spencer (1979). "Torsion algebraic cycles and a theorem of Roitman". Compositio Mathematica . 39 (1). MR   0539002.
  3. Milne, J. S. (1982). "Zero cycles on algebraic varieties in nonzero characteristic : Rojtman's theorem". Compositio Mathematica. 47 (3): 271–287.
  4. Spieß, Michael; Szamuely, Tamás (2003). "On the Albanese map for smooth quasi-projective varieties". Mathematische Annalen . 325: 1–17. arXiv: math/0009017 . doi:10.1007/s00208-002-0359-8. S2CID   14014858.
  5. Geisser, Thomas (2015). "Rojtman's theorem for normal schemes". Mathematical Research Letters. 22 (4): 1129–1144. arXiv: 1402.1831 . doi:10.4310/MRL.2015.v22.n4.a8. S2CID   59423465.

Related Research Articles

In mathematics, the Hodge conjecture is a major unsolved problem in algebraic geometry and complex geometry that relates the algebraic topology of a non-singular complex algebraic variety to its subvarieties.

Abelian variety

In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions. Abelian varieties are at the same time among the most studied objects in algebraic geometry and indispensable tools for much research on other topics in algebraic geometry and number theory.

Algebraic group Algebraic variety with a group structure

In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory.

Algebraic variety Mathematical object studied in the field of algebraic geometry

Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition.

Projective variety

In algebraic geometry, a projective variety over an algebraically closed field k is a subset of some projective n-space over k that is the zero-locus of some finite family of homogeneous polynomials of n + 1 variables with coefficients in k, that generate a prime ideal, the defining ideal of the variety. Equivalently, an algebraic variety is projective if it can be embedded as a Zariski closed subvariety of .

Linear algebraic group Subgroup of the group of invertible n×n matrices

In mathematics, a linear algebraic group is a subgroup of the group of invertible matrices that is defined by polynomial equations. An example is the orthogonal group, defined by the relation where is the transpose of .

In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures. Étale cohomology theory can be used to construct ℓ-adic cohomology, which is an example of a Weil cohomology theory in algebraic geometry. This has many applications, such as the proof of the Weil conjectures and the construction of representations of finite groups of Lie type.

In algebraic geometry, motives is a theory proposed by Alexander Grothendieck in the 1960s to unify the vast array of similarly behaved cohomology theories such as singular cohomology, de Rham cohomology, etale cohomology, and crystalline cohomology. Philosophically, a "motif" is the "cohomology essence" of a variety.

Algebraic K-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called K-groups. These are groups in the sense of abstract algebra. They contain detailed information about the original object but are notoriously difficult to compute; for example, an important outstanding problem is to compute the K-groups of the integers.

K3 surface Type of smooth complex surface of kodaira dimension 0

In mathematics, a complex analytic K3 surface is a compact connected complex manifold of dimension 2 with trivial canonical bundle and irregularity zero. An (algebraic) K3 surface over any field means a smooth proper geometrically connected algebraic surface that satisfies the same conditions. In the Enriques–Kodaira classification of surfaces, K3 surfaces form one of the four classes of minimal surfaces of Kodaira dimension zero. A simple example is the Fermat quartic surface

In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors and Weil divisors. Both are derived from the notion of divisibility in the integers and algebraic number fields.

In mathematics, the Picard group of a ringed space X, denoted by Pic(X), is the group of isomorphism classes of invertible sheaves on X, with the group operation being tensor product. This construction is a global version of the construction of the divisor class group, or ideal class group, and is much used in algebraic geometry and the theory of complex manifolds.

In mathematics, the Schottky problem, named after Friedrich Schottky, is a classical question of algebraic geometry, asking for a characterisation of Jacobian varieties amongst abelian varieties.

In arithmetic geometry, the Mordell–Weil group is an abelian group associated to any abelian variety defined over a number field , it is an arithmetic invariant of the Abelian variety. It is simply the group of -points of , so is the Mordell–Weil grouppg 207. The main structure theorem about this group is the Mordell–Weil theorem which shows this group is in fact a finitely-generated abelian group. Moreover, there are many conjectures related to this group, such as the Birch and Swinnerton-Dyer conjecture which relates the rank of to the zero of the associated L-function at a special point.

In mathematics, a dual abelian variety can be defined from an abelian variety A, defined over a field K.

In mathematics, an algebraic cycle on an algebraic variety V is a formal linear combination of subvarieties of V. These are the part of the algebraic topology of V that is directly accessible by algebraic methods. Understanding the algebraic cycles on a variety can give profound insights into the structure of the variety.

In mathematics, the Abel–Jacobi map is a construction of algebraic geometry which relates an algebraic curve to its Jacobian variety. In Riemannian geometry, it is a more general construction mapping a manifold to its Jacobi torus. The name derives from the theorem of Abel and Jacobi that two effective divisors are linearly equivalent if and only if they are indistinguishable under the Abel–Jacobi map.

In mathematics, the irregularity of a complex surface X is the Hodge number , usually denoted by q. The irregularity of an algebraic surface is sometimes defined to be this Hodge number, and sometimes defined to be the dimension of the Picard variety, which is the same in characteristic 0 but can be smaller in positive characteristic.

This is a glossary of algebraic geometry.