  We prove the vanishing of the Dolbeault cohomology groups on Hermitian
manifolds with $dd^c$-harmonic K\"ahler form and positive (1,1)-part of the
Ricci form of the Bismut connection. This implies the vanishing of the
Dolbeault cohomology groups on complex surfaces which admit a conformal class
of Hermitian metrics, such that the Ricci tensor of the canonical Weyl
structure is positive. As a corollary we obtain that any such surface must be
rational with $c_1^2 >0$. As an application, the pth Dolbeault cohomology
groups of a left-invariant complex structure compatible with a bi-invariant
metric on a compact even dimensional Lie group are computed.
