Matsushima criterion

The homogeneous space $ G/H $ , where $ G $ is an affine reductive algebraic group (cf. also Affine group; Reductive group) defined over an algebraically closed field $ k $ and $ H $ is a closed subgroup of $ G $ , is an affine algebraic variety if and only if $ H $ is a reductive group. This criterion was first found by Y. Matsushima [1] in the case where $ k $ is the complex field. Later, proofs were given that are valid for every algebraically closed field of characteristic zero (see [2], [3], [4]). In the case where the characteristic of $ k $ is positive, the proof of the criterion was obtained only after the proof of the Mumford hypothesis (see [5], [6]).

References

[1]	Y. Matsushima, "Espaces homogènes de Stein des groupes de Lie complexes" Nagoya Math. J. , 16 (1960) pp. 205–218 MR0109854 Zbl 0094.28201
[2]	A. Białynicki-Birula, "On homogeneous affine spaces of linear algebraic groups" Amer. J. Math. , 85 (1963) pp. 577–582 Zbl 0116.38202
[3]	D. Luna, "Slices étales" Bull. Soc. Math. France , 33 (1973) pp. 81–105 MR0342523 Zbl 0286.14014
[4]	A. Borel, Harish-Chandra, "Arithmetic subgroups of algebraic groups" Ann. of Math. , 75 (1962) pp. 485–535 MR0147566 Zbl 0107.14804
[5]	E.A. Nisnevich, "Affine homogeneous spaces and finite subgroups of arithmetic groups over function fields" Funct. Anal. Appl. , 11 : 1 (1977) pp. 64–65 Funktsional. Anal. i Prilozhen. , 11 : 1 (1977) pp. 73–74
[6]	R.W. Richardson, "Affine coset spaces of reductive algebraic groups" Bull. London Math. Soc. , 9 (1977) pp. 38–41 MR0437549 Zbl 0355.14020