The mean curvature of transverse K\"ahler foliations

TL;DR

This paper investigates the properties of the mean curvature one-form in transverse Kähler foliations, revealing restrictions on basic cohomology akin to Kähler manifolds, especially under automorphic mean curvature conditions.

Contribution

It establishes new restrictions on basic cohomology for transverse Kähler foliations with automorphic mean curvature, extending Kähler geometry concepts to foliations.

Findings

Odd basic Betti numbers are even under automorphic mean curvature.

Full Hodge diamond structure is absent unless the foliation is taut.

Restrictions resemble those in classical Kähler geometry.

Abstract

We study properties of the mean curvature one-form and its holomorphic and antiholomorphic cousins on a transverse K\"ahler foliation. If the mean curvature of the foliation is automorphic, then there are some restrictions on basic cohomology similar to that on K\"ahler manifolds, such as the requirement that the odd basic Betti numbers must be even. However, the full Hodge diamond structure does not apply to basic Dolbeault cohomology unless the foliation is taut.