Abstract
Complete smooth complex algebraic varieties with an almost transitive action of a linear algebraic group are studied. They are classified in the case, when the complement of the open orbit is a homogeneous hypersurface. If the group and the isotropy subgroup at a generic point are both reductive, then there exists a natural one-to-one correspondence between these two-orbit varieties and compact riemannian symmetric spaces of rank one.
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Ahiezer, D. Equivariant completions of homogenous algebraic varieties by homogenous divisors. Ann Glob Anal Geom 1, 49–78 (1983). https://doi.org/10.1007/BF02329739
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DOI: https://doi.org/10.1007/BF02329739