Smooth foliations on homogeneous compact K\"ahler manifolds

TL;DR

This paper investigates smooth foliations on homogeneous compact K"ahler manifolds, proving local triviality for rational cases and classifying those with dense leaves, based on a structure theorem and Borel-Remmert decomposition.

Contribution

It provides a classification of smooth foliations on homogeneous compact K"ahler manifolds and establishes local triviality for rational cases, advancing understanding of foliation structures.

Findings

Smooth foliations on rational compact homogeneous manifolds are locally trivial fibrations.

Classification of smooth foliations with all leaves analytically dense.

A structure theorem relating foliations to Borel-Remmert decomposition.

Abstract

We study smooth foliations of arbitrary codimension on homogeneous compact K\"ahler manifolds. We prove that smooth foliations on rational compact homogeneous manifolds are locally trivial fibrations and classify the smooth foliations with all leaves analytically dense on compact homogeneous K\"ahler manifolds. Both results are builded upon a (rough) structure Theorem for smooth foliations on compact homogeneous K\"ahler manifolds obtained by comparison of the foliation and the Borel-Remmert decomposition of the ambient.