Complete homogeneous varieties: Structure and classification

Sancho de Salas, Carlos

doi:10.1090/S0002-9947-03-03280-X

Complete homogeneous varieties: Structure and classification
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by Carlos Sancho de Salas PDF

Trans. Amer. Math. Soc. 355 (2003), 3651-3667 Request permission

Abstract:

Homogeneous varieties are those whose group of automorphisms acts transitively on them. In this paper we prove that any complete homogeneous variety splits in a unique way as a product of an abelian variety and a parabolic variety. This is obtained by proving a rigidity theorem for the parabolic subgroups of a linear group. Finally, using the results of Wenzel on the classification of parabolic subgroups of a linear group and the results of Demazure on the automorphisms of a flag variety, we obtain the classification of the parabolic varieties (in characteristic different from $2,3$). This, together with the moduli of abelian varieties, concludes the classification of the complete homogeneous varieties.

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