Albanese variety

# Albanese variety

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In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the Albanese variety A(V), named for Giacomo Albanese
Giacomo Albanese
Giacomo Albanese was an Italian mathematician known for his work in algebraic geometry. He took a permanent position in São Paulo, Brazil, in 1936.-External links:...

, is a generalization of the Jacobian variety
Jacobian variety
In mathematics, the Jacobian variety J of a non-singular algebraic curve C of genus g is the moduli space of degree 0 line bundles...

of a curve, and is the abelian variety generated by a variety V. 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 factors uniquely through A(V). For complex manifolds 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. (Although it is called a variety in this case, it need not be algebraic.)

For compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...

Kähler manifold
Kähler manifold
In mathematics, a Kähler manifold is a manifold with unitary structure satisfying an integrability condition.In particular, it is a Riemannian manifold, a complex manifold, and a symplectic manifold, with these three structures all mutually compatible.This threefold structure corresponds to the...

s 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
Irregularity of a surface
In mathematics, the irregularity of a complex surface X is the Hodge number h0,1= dim H1, 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...

. In terms of differential form
Differential form
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...

s, any holomorphic 1-form on V is a pullback of an invariant 1-form on the Albanese, coming from the holomorphic cotangent space
Cotangent space
In differential geometry, one can attach to every point x of a smooth manifold a vector space called the cotangent space at x. Typically, the cotangent space is defined as the dual space of the tangent space at x, although there are more direct definitions...

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).

## Connection to Picard variety

The Albanese variety is dual to the Picard variety (the connected component
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...

of zero of the Picard scheme classifying invertible sheaves on V):