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

Abstract

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 Clifford–Klein 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 Clifford–Klein forms.