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The homogeneous space $G / H$, where $k$ is an affine reductive [[Algebraic group|algebraic group]] (cf. also [[Affine group|Affine group]]; [[Reductive group|Reductive group]]) defined over an algebraically closed field $k$ and $H$ is a closed subgroup of $k$, is an affine algebraic variety if and only if $H$ is a reductive group. This criterion was first found by Y. Matsushima [[#References|[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 [[#References|[2]]], [[#References|[3]]], [[#References|[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|Mumford hypothesis]] (see [[#References|[5]]], [[#References|[6]]]).

====References====
<table><tr><td valign="top">[1]</td> <td valign="top"> Y. Matsushima, "Espaces homogènes de Stein des groupes de Lie complexes" ''Nagoya Math. J.'' , '''16''' (1960) pp. 205–218 {{MR|0109854}} {{ZBL|0094.28201}} </td></tr><tr><td valign="top">[2]</td> <td valign="top"> A. Białynicki-Birula, "On homogeneous affine spaces of linear algebraic groups" ''Amer. J. Math.'' , '''85''' (1963) pp. 577–582 {{MR|}} {{ZBL|0116.38202}} </td></tr><tr><td valign="top">[3]</td> <td valign="top"> D. Luna, "Slices étales" ''Bull. Soc. Math. France'' , '''33''' (1973) pp. 81–105 {{MR|0342523}} {{ZBL|0286.14014}} </td></tr><tr><td valign="top">[4]</td> <td valign="top"> A. Borel, Harish-Chandra, "Arithmetic subgroups of algebraic groups" ''Ann. of Math.'' , '''75''' (1962) pp. 485–535 {{MR|0147566}} {{ZBL|0107.14804}} </td></tr><tr><td valign="top">[5]</td> <td valign="top"> 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 {{MR|}} {{ZBL|}} </td></tr><tr><td valign="top">[6]</td> <td valign="top"> R.W. Richardson, "Affine coset spaces of reductive algebraic groups" ''Bull. London Math. Soc.'' , '''9''' (1977) pp. 38–41 {{MR|0437549}} {{ZBL|0355.14020}} </td></tr></table>