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.Comment: 30 pages, revised notation section and historical information so that
there is not much overlap with previous pape
