Vanishing theorems on complete Kähler manifolds
The author proves the following \(L^ 2\)-vanishing theorem. Let X be a complete Kähler manifold of dimension n, let (E,h) be a hermitian bundle over X, and let \(\sigma\) be a d-closed semipositive (1,1)-form on X. Assume that the curvature form for h is equal to or greater than \(\sigma\). Then, for any \({\bar \sigma}\)-closed E-valued (n,q)-form f which is square integrable with respect to \(\sigma\), we can find an E- valued (n,q-1)-form g which is square integrable with respect to \(\sigma\) satisfying \({\bar \partial}g=f\). Here \(q\geq 1\). - Then the author applies the above theorem to obtain the following two vanishing theorems: Let X be a compact Kähler manifold, let Y be an analytic space, let \(f: X\to Y\) be a holomorphic map, and let (E,h) be a hermitian bundle over X. Assume that the curvature form h is equal to or greater than the pull- back of a Kähler metric on Y. Then, \(H^ q(Y,f_*{\mathcal O}(K_ X\otimes E))=0\) for \(q\geq 1\). - Here \(K_ X\) denotes the canonical bundle of X and \(f_*{\mathcal O}(K_ X\otimes E)\) denotes the direct image sheaf of \({\mathcal O}(K_ X\otimes E).\) Theorem. Let X be a 1-convex manifold with maximal compact analytic set A, and let \(E\to X\) be a holomorphic vector bundle. Assume the restriction of E to A is Nakano-semipositive. Then \(H^ q(X,0(K_ X\otimes E))=0\) for \(q\geq 1\). - Several interesting applications of the above results are given.
scientific article; zbMATH DE number 3906869