- Albanese variety
mathematics, the Albanese variety is a construction of algebraic geometry, which for an algebraic variety"V" solves a universal problemfor morphisms of "V" into abelian varieties; it is the " abelianization" of a variety, and expresses abelian varieties as a reflective subcategoryof algebraic varieties.
It is dual to (the identity component of) the
In the classical case of complex projective
non-singularvarieties, the Albanese variety "Alb"("V") is a complex torusconstructed from "V", of (complex) dimension the Hodge number"h"0,1, that is, the dimension of the space of differentials of the first kindon "V". The construction is named for Giacomo Albanese.
The Albanese variety generalises the construction of the
Jacobian varietyof 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 spaceof "Alb"("V") at its identity element. Just as for the curve case, by choice of a base pointon "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 varietyclassifying invertible sheaveson "V", and this defines it. The duality theory of abelian varietiesis used to pass from the Picard variety, which is constructed as a representable functor, to the Albanese.
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