Vanishing theorems on Hermitian manifolds

✍ Scribed by Bogdan Alexandrov; Stefan Ivanov

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 107 KB
• Volume: 14
• Category: Article
• ISSN: 0926-2245
• DOI: 10.1016/s0926-2245(01)00044-4

✦ Synopsis

We prove the vanishing of the Dolbeault cohomology groups on Hermitian manifolds with dd charmonic Kähler 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 2 1 > 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.