Invariant holomorphic foliations on Kobayashi hyperbolic homogeneous manifolds

TL;DR

This paper studies invariant holomorphic foliations on Kobayashi hyperbolic homogeneous manifolds, showing they are fibers of a holomorphic submersion, and analyzes fixed point sets of automorphism families on hyperbolic convex domains.

Contribution

It proves that invariant holomorphic foliations are fibers of a G-equivariant submersion and characterizes fixed point sets of automorphism families on hyperbolic convex domains.

Findings

Leaves of invariant foliations are fibers of a G-equivariant submersion.

Fixed point sets of automorphism families are either empty or connected complex submanifolds.

Provides structure results for automorphisms on hyperbolic convex domains.

Abstract

Let $M$ be a Kobayashi hyperbolic homogenous manifold. Let $\mathcal F$ be a holomorphic foliation on $M$ invariant under a transitive group $G$ of biholomorphisms. We prove that the leaves of $\mathcal F$ are the fibers of a holomorphic $G$-equivariant submersion $\pi \colon M \to N$ onto a $G$-homogeneous complex manifold $N$. We also show that if $\mathcal Q$ is an automorphism family of a hyperbolic convex (possibly unbounded) domain $D$ in $\mathbb C^n$, then the fixed point set of $\mathcal Q$ is either empty or a connected complex submanifold of $D$.