On homogeneity of two-orbit varieties with respect to the full group of automorphisms. (Russian) Zbl 0698.14054

Akhiezer, D. N.

Vopr. Teor. Grupp Gomologicheskoj Algebry 8, 141-154 (1988).

Theorem: Let X be a complete non-singular complex algebraic variety, and let G be a connected linear algebraic group which acts regularly on X. Let X be the union of two orbits of group G and let the closed orbit A have codimension 1. Then the following conditions are equivalent:
(a) \(H^ 0(A,{\mathbb{N}}_{X| A})\neq 0\), where \({\mathbb{N}}_{X| A}\) is the normal bundle to A in X;
(b) X is homogeneous under Aut(X);
(c) \(X=X_ 1\times X_ 2\) and the image of G in Aut(X) is a direct product \(G_ 1\times G_ 2\), where \(G_ 1\) acts transitively on \(X_ 1\), \(G_ 2\) acts transitively on \(X_ 2\) and some alternative formulated in the paper takes place.

Reviewer: N.I.Osetinski

MSC:

14L30	Group actions on varieties or schemes (quotients)
14M17	Homogeneous spaces and generalizations

Keywords:

action of linear group; automorphism group