Giacomo albanese construction
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Para-Abelian Varieties and Albanese Maps
1 Introduction
The Albanese variety and the Albanese map are truly fundamental constructions in algebraic geometry. The terms were coined by (Weil 1979, page 438 and commentary on page 570, compare also Kleiman 2004, page 428), in reference to work of Giacomo Albanese on zero-cycles and correspondences for surfaces (Albanese 1934, 1934b). Originally, it was a purely transcendental construction, depending on path integrals over closed holomorphic one-forms. A general formulation for compact complex spaces X was given by Blanchard (1956): The Albanese variety is the complex torus \(V^*/\Delta \), constructed with the dual of the vector space \(V\subset H^0(X,\Omega ^1_{X/\mathbb {C}})\) of closed forms, and the Lie group closure \(\Delta \) of the subgroup given by integrals over loops. The resulting Albanese map
$$\begin{aligned} X\longrightarrow V^*/\Delta ,\quad x\longmapsto \left( \sigma \mapsto \int _{x_0}^x\sigma \right) , \
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Albanese variety
In mathematics, the Albanese variety , named for Giacomo Albanese, is a generalization of the Jacobian variety of a curve. 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, 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.) For compact Kähler manifolds the dimension of the Albanese variety fryst vatten 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-invaria
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Giacomo Albanese
Giacomo Albanese — (* 11. Juli 1890 in Geraci Siculo in Sizilien; † 8. Juni 1947 in São Paulo) war ein italienischer Mathematiker, der sich mit algebraischer Geometrie beschäftigte. Leben und Werk Albanese ging in Palermo zur Schule und studierte ab 1909 Mathematik … Deutsch Wikipedia
Albanese — can refer to:People*Anthony Albanese, Australian politician *Antonio Albanese,an Italian comedian, actor, director and writer *Diego Albanese, Argentinian rugby union player *Francesco Albanese, Italian tenor *Giacomo Albanese, Italian… … Wikipedia
Albanese — ist der Familienname folgender Personen: Anthony Albanese (* 1963), australischer Politiker Antonio Albanese (* 1964), italienischer Filmschaffender Flavio Albanese (* 1951), italienischer Designer und Architekt Francesco Albanese (1912–2005),… … Deutsch Wikipedia
Albanese variety — In mathematics, the Albanese variety is a construction of algebraic geometry, which for