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Smooth foliations on homogeneous compact K\"ahler manifolds

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.