Vanishing theorems on Hermitian manifolds (Q5946127)

The authors prove that for every \(2n\)-dimensional (\(n\geq 2\)) compact Hermitian manifold \(( M,g,J) \) with Bismut connection (the Hermitian connection with totally skew-symmetric torsion) and \(dd^c\)-harmonic Kähler form \(\Omega \) satisfying the condition that the \(( 1,1) \)-part of the associated Ricci form on \(M\) is nonnegative everywhere and positive somewhere, the Dolbeault cohomology \(p\)-groups vanish for \(p=1,\dots ,n.\) There is a consequent result for compact complex surfaces admitting a conformal class of Hermitian metrics such that the symmetric part of the Ricci tensor of the canonical Weyl structure is nonnegative everywhere and positive somewhere. If \(( M,g,J) \) with Gauduchon metric \(g\) is locally conformally Kähler and the \(( 1,1) \)-part of the Ricci form is nonnegative on \(M\), then for even \(n>2\) either the Dolbeault cohomology \(p\)-group vanishes for \(p=n/2\) or \(M\) is Kähler, with further results for the class of generalized Hopf manifolds.