RATIONALITY OF HOMOGENEOUS VARIETIES

Abstract

Let G be a connected linear algebraic group over an algebraically closed field k, and let H be a connected closed subgroup of G. We prove that the homogeneous variety G/H is a rational variety over k whenever H is solvable or when dim(G/H) ≤ 10 and char(k) = 0. When H is of maximal rank in G, we also prove that G/H is rational if the maximal semisimple quotient of G is isogenous to a product of almost-simple groups of type A, type C (when char(k) ≠ 2), or type B3 or G2 (when char(k) = 0).