On odd dimensional complex analytic Kleinian groups

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.