Vanishing theorems on Hermitian manifolds

We prove the vanishing of the Dolbeault cohomology groups on Hermitian manifolds with dd(c)-harmonic Kahler 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 bi-invariant metric on a compact even dimensional Lie group are computed.