# On the Hirzebruch-Kobayashi-Ono proportionality principle and the non-existence of compact solvable Clifford-Klein forms of certain homogeneous spaces

## TL;DR

This paper proves that most symmetric and 3-symmetric spaces cannot have compact solvable Clifford-Klein forms, advancing understanding of the geometric structures of homogeneous spaces.

## Contribution

It introduces a new sufficient condition for the non-existence of compact solvable Clifford-Klein forms using the Hirzebruch-Kobayashi-Ono principle and syndetic hulls theory.

## Key findings

- Most symmetric spaces do not admit such forms.
- The method applies to a broad class of homogeneous spaces.
- Several potential exceptions are identified.

## Abstract

This article continues a line of research aimed at solving an important problem of T. Kobayashi of the existence of compact Clifford-Klein forms of reductive homogeneous spaces. We contribute to this topic by showing that almost all symmetric spaces and 3-symmetric spaces do not admit solvable compact CliffordfKlein forms (with several possible exceptions). Our basic tool is a combination of the Hirzebruch-Kobayashi-Ono proportionality principle with the theory of syndetic hulls. Using this, we prove a general theorem which yields a sufficient condition for the non-existence of compact solvable CliffordKlein forms.