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Some basic results on actions of nonaffine algebraic groups. (English) Zbl 1217.14029
Campbell, H.E.A. (ed.) et al., Symmetry and spaces. In Honor of Gerry Schwarz on the occasion of his 60th birthday. Basel: Birkhäuser (ISBN 978-0-8176-4874-9/hbk; 978-0-8176-4875-6/ebook). Progress in Mathematics 278, 1-20 (2010).
In this article, the author studies actions of connected nonaffine algebraic groups on normal varieties. Algebraic group actions of affine groups have been extensively studied, however, little is known about the case of nonaffine groups. The author shows that nonaffine group actions can be reduced to actions of affine subgroup schemes, in the case of actions on normal varieties. It is shown that any normal $$G$$-variety admits an open cover by $$G$$-stable quasi-projective varieties. It is also shown that any normal quasi-projective variety on which $$G$$ acts faithfully admits a $$G$$-equivariant embedding into the projectivization of a $$G$$-homogeneous vector bundle over a certain variety. These results generalize theorems of Sumihiro for the case when $$G$$ is affine.
For the entire collection see [Zbl 1182.14002].

##### MSC:
 14J50 Automorphisms of surfaces and higher-dimensional varieties 14L30 Group actions on varieties or schemes (quotients) 14M17 Homogeneous spaces and generalizations
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