Vanishing theorems on Hermitian manifolds
نویسنده
چکیده
We prove the vanishing of the Dolbeault cohomology groups on Hermitian manifolds with ddc-harmonic 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 1 > 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. Running title: Vanishing theorems on Hermitian manifolds
منابع مشابه
Vanishing Theorems and String Backgrounds
We show various vanishing theorems for the cohomology groups of compact hermitian man-ifolds for which the Bismut connection has (restricted) holonomy contained in SU (n) and classify all such manifolds of dimension four. In this way we provide necessary conditions for the existence of such structures on hermitian manifolds. Then we apply our results to solutions of the string equations and sho...
متن کاملThe Bochner identities for the Kählerian gradients
We discuss algebraic properties for the symbols of geometric first order differential operators on almost Hermitian manifolds and Kähler manifolds. Through study on the universal enveloping algebra and higher Casimir elements, we know algebraic relations for the symbols like the Clifford algebra. From the relations, we have all the Bochner identities for the operators. As applications, we have ...
متن کاملVanishing Theorems on Complete K Ahler Manifolds and Their Applications
Semi-positive line bundles over compact Kahler manifolds have been the focus of studies for decades. Among them, there are several straddling vanishing theorems such as the Kodaira-Nakano Vanishing Theorem, Vesentini-Gigante-Girbau Vanishing Theorems and KawamataViehweg Vanishing Theorem. As a corollary of the above mentioned vanishing theorems one can easily show that a line bundle over compa...
متن کاملVanishing Theorems on Covering Manifolds
Let M be an oriented even-dimensional Riemannian manifold on which a discrete group Γ of orientation-preserving isometries acts freely, so that the quotientX = M/Γ is compact. We prove a vanishing theorem for a half-kernel of a Γ-invariant Dirac operator on a Γ-equivariant Clifford module overM , twisted by a sufficiently large power of a Γ-equivariant line bundle, whose curvature is non-degene...
متن کاملHiggs Bundles and Holomorphic Forms
For a complex manifold X which has a holomorphic form ̟ of odd degree k, we endow E = ⊕ p≥a Λ (p,0)(X) with a Higgs bundle structure θ given by θ(Z)(φ) := {i(Z)̟} ∧ φ. The properties such as curvature and stability of these and other Higgs bundles are examined. We prove (Theorem 2, section 2, for k > 1) E and additional classes of Higgs subbundles of E do not admit Higgs-Hermitian-Yang-Mills metr...
متن کامل