Numerically flat foliations and holomorphic Poisson geometry

TL;DR

This paper studies the structure of holomorphic foliations with flat tangent bundles on compact Kähler manifolds, revealing their decomposition, leaf uniformization, and implications for Poisson geometry and Hodge theory.

Contribution

It extends previous results to show that such foliations induce tangent bundle decompositions and have leaves modeled on Euclidean spaces, with applications to Poisson structures.

Findings

Foliations induce a decomposition of the tangent bundle.

Leaves are uniformized by Euclidean spaces.

Global Weinstein splitting theorem for rank-two Poisson structures.

Abstract

We investigate the structure of smooth holomorphic foliations with numerically flat tangent bundles on compact K\"ahler manifolds. Extending earlier results on non-uniruled projective manifolds by the second and fourth authors, we show that such foliations induce a decomposition of the tangent bundle of the ambient manifold, have leaves uniformized by Euclidean spaces, and have torsion canonical bundle. Additionally, we prove that smooth two-dimensional foliations with numerically trivial canonical bundle on projective manifolds are either isotrivial fibrations or have numerically flat tangent bundles. This in turn implies a global Weinstein splitting theorem for rank-two Poisson structures on projective manifolds. We also derive new Hodge-theoretic conditions for the existence of zeros of Poisson structures on compact K\"ahler manifolds.