# Albanese variety

Albanese variety

In mathematics, the Albanese variety is a construction of algebraic geometry, which for an algebraic variety "V" solves a universal problem for morphisms of "V" into abelian varieties; it is the "abelianization" of a variety, and expresses abelian varieties as a reflective subcategory of algebraic varieties.

It is dual to (the identity component of) the Picard variety::$operatorname\left\{Alb\right\},V = \left(operatorname\left\{Pic\right\}_0,V\right)^*$

In the classical case of complex projective non-singular varieties, the Albanese variety "Alb"("V") is a complex torus constructed from "V", of (complex) dimension the Hodge number "h"0,1, that is, the dimension of the space of differentials of the first kind on "V". The construction is named for Giacomo Albanese.

The Albanese variety generalises the construction of the Jacobian variety of an algebraic curve; and was introduced to study algebraic surfaces. There the dimension of the Albanese is also the number "h"1,0, traditionally called the "irregularity" of a surface. In terms of differential forms, any holomorphic 1-form on "V" is a pullback of an 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

:"V" → "Alb"("V")

is defined, along which the 1-forms pull back. This morphism is well-defined only up to a translation on the Albanese.

Connection to Picard variety

The Albanese variety is dual to the (connected component of zero of the) Picard variety classifying invertible sheaves on "V", and this defines it. The duality theory of abelian varieties is used to pass from the Picard variety, which is constructed as a representable functor, to the Albanese.

References

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