## Journal of the Mathematical Society of Japan

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

Yosuke MORITA

#### 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 $\operatorname{rank} G - \operatorname{rank} K < \operatorname{rank} H - \operatorname{rank} K_H$. 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.

#### Note

This work was supported by JSPS KAKENHI Grant Numbers 14J08233 and 17H06784, and the Program for Leading Graduate Schools, MEXT, Japan.

#### Article information

Source
J. Math. Soc. Japan, Advance publication (2019), 19 pages.

Dates
Revised: 29 May 2018
First available in Project Euclid: 3 July 2019

https://projecteuclid.org/euclid.jmsj/1562140845

Digital Object Identifier
doi:10.2969/jmsj/78007800

#### Citation

MORITA, Yosuke. Proof of Kobayashi's rank conjecture on Clifford–Klein forms. J. Math. Soc. Japan, advance publication, 3 July 2019. doi:10.2969/jmsj/78007800. https://projecteuclid.org/euclid.jmsj/1562140845

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