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

Abstract

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.