  We describe the basic Dolbealut cohomology algebra of the canonical foliation
on a class of complex manifolds with a torus symmetry group. This class
includes complex moment-angle manifolds, LVM- and LVMB-manifolds and, in most
generality, complex manifolds with a maximal holomorphic torus action. We also
provide a dga model for the ordinary Dolbeault cohomology algebra. The Hodge
decomposition for the basic Dolbeault cohomology is proved by reducing to the
transversely Kaehler (equivalently, polytopal) case using a foliated analogue
of toric blow-up.
