The mean curvature of transverse K\"ahler foliations

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.