Proof of Kobayashi's rank conjecture on Clifford–Klein forms

Abstract

T. Kobayashi conjectured in the 36th Geometry Symposium in Japan (1989) that a homogeneous space G/H of reductive type does not admit a compact Clifford-Klein form if rank G - rank K < rank H - rank K_H. We solve this conjecture affirmatively. We apply a cohomological obstruction to the existence of compact Clifford-Klein forms proved previously by the author, and use the Sullivan model for a reductive pair due to Cartan-Chevalley-Koszul-Weil.