On odd dimensional complex analytic Kleinian groups

TL;DR

This paper proposes a generalization of complex analytic Kleinian groups to odd dimensions, defining domains of discontinuity for certain subgroups of PGL_{2n+1}(\u00a3) and exploring their quotients, which include interesting non-Ke4hler manifolds.

Contribution

It introduces a new framework for higher-dimensional Kleinian groups in odd dimensions, extending classical theory and constructing novel non-Ke4hler manifolds as quotients.

Findings

Defined canonical domains of discontinuity for odd-dimensional cases.

Constructed examples of non-Ke4hler manifolds as quotients.

Extended classical Kleinian group theory to higher odd dimensions.

Abstract

We shall explain here an idea to generalize classical complex analytic Kleinian group theory to any odd dimensional cases. For a certain class of discrete subgroups of $\PGL_{2n+1}(\C)$ acting on $\P^{2n+1}$, we can define their domains of discontinuity in a canonical manner, regarding an $n$-dimensional projective linear subspace in $\P^{2n+1}$ as a point, like a point in the classical $1$-dimensional case. Many interesting (compact) non-K\"ahler manifolds appear systematically as the canonical quotients of the domains. In the last section, we shall give some examples.