Self-intersection of foliated cycles on complex manifolds

TL;DR

This paper proves that on compact Kahler manifolds, certain foliated cycles have zero self-intersection unless they are supported on compact manifolds, with implications for the structure of laminations in projective spaces.

Contribution

It establishes the vanishing of self-intersection for foliated cycles under specific conditions and characterizes the foliated cycles in low codimension laminations.

Findings

Self-intersection of the cohomology class of T vanishes unless T contains currents of integration along compact manifolds.

Transversally Lipschitz laminations of low codimension in projective spaces do not carry non-trivial foliated cycles.

The result constrains the structure of foliated cycles in complex manifolds with lamination structures.

Abstract

Let X be a compact Kahler manifold and let T be a foliated cycle directed by a transversally Lipschitz lamination on X . We prove that the self-intersection of the cohomology class of T vanishes as long as T does not contain currents of integration along compact manifolds. As a consequence we prove that transversally Lipschitz laminations of low codimension in certain manifolds, e.g. projective spaces, do not carry any foliated cycles except those given by integration along compact leaves.