- Albanese variety
In

mathematics , the**Albanese variety**is a construction ofalgebraic geometry , which for analgebraic variety "V" solves auniversal problem formorphism s of "V" intoabelian varieties ; it is the "abelianization " of a variety, and expresses abelian varieties as areflective subcategory of algebraic varieties.It is dual to (the identity component of) the

Picard variety ::$operatorname\{Alb\},V\; =\; (operatorname\{Pic\}\_0,V)^*$In the classical case of complex projective

non-singular varieties, the Albanese variety "Alb"("V") is a complextorus constructed from "V", of (complex) dimension theHodge number "h"^{0,1}, that is, the dimension of the space ofdifferentials of the first kind on "V". The construction is named forGiacomo Albanese .The Albanese variety generalises the construction of the

Jacobian variety of analgebraic curve ; and was introduced to studyalgebraic surface s. There the dimension of the Albanese is also the number "h"^{1,0}, traditionally called the "irregularity" of a surface. In terms ofdifferential form s, any holomorphic 1-form on "V" is a pullback of an invariant 1-form on the Albanese, coming from the holomorphiccotangent space of "Alb"("V") at its identity element. Just as for the curve case, by choice of abase 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 classifyinginvertible sheaves on "V", and this defines it. Theduality theory of abelian varieties is used to pass from the Picard variety, which is constructed as arepresentable functor , to the Albanese.See also:

Roitman's theorem .**References***

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