# Proof of Kobayashi's rank conjecture on Clifford-Klein forms

## TL;DR

This paper proves Kobayashi's conjecture that certain homogeneous spaces do not admit compact Clifford-Klein forms when a specific rank inequality holds, using cohomological obstructions and Sullivan models.

## Contribution

It provides a complete proof of Kobayashi's rank conjecture using advanced cohomological techniques and Sullivan models for reductive pairs.

## Key findings

- Confirmed Kobayashi's conjecture affirmatively.
- Established a cohomological obstruction criterion.
- Applied Sullivan models to the problem.

## 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.