Journal of the Mathematical Society of Japan

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

Yosuke MORITA

Advance publication

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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
Received: 18 May 2017
Revised: 29 May 2018
First available in Project Euclid: 3 July 2019

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1562140845

Digital Object Identifier
doi:10.2969/jmsj/78007800

Subjects
Primary: 57S30: Discontinuous groups of transformations
Secondary: 17B56: Cohomology of Lie (super)algebras 55P62: Rational homotopy theory 57T15: Homology and cohomology of homogeneous spaces of Lie groups

Keywords
Clifford–Klein form pure Sullivan algebra relative Lie algebra cohomology transgression

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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