Vanishing theorems on Hermitian manifolds

We prove the vanishing of the Dolbeault cohomology groups on Hermitian manifolds with ddc-harmonic K?hler 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 c12>0. As an application, the pth Dolbeault cohomology groups of a left-invariant complex structure compatible with bi-invariant metric on a compact even dimensional Lie group are computed.