Vanishing theorems for tensor powers of an ample vector bundle
Let X be a compact complex manifold of dimension n and E resp. L an ample holomorphic vector bundle of rank r resp. an ample line bundle on X. The paper gives generalizations of Griffiths' vanishing theorem \(H^{n,q}(X,S^ kE\otimes \det (E\otimes L))=0\) for \(q\geq 1\) [\textit{P. A. Griffiths}, Global Analysis, papers in Honor of K. Kodaira, 185-251 (1969; Zbl 0201.240)] which shall not be repeated here and Le Potier's vanishing theorem \(H^{p,q}(X,E)=0\) for \(p+q\geq n+r\) [\textit{J. Le Potier}, Math. Ann. 218, 35-53 (1975; Zbl 0313.32037)] saying that \(H^{p,q}(X,E^{\otimes k}\otimes (\det (E))^{\ell}\otimes L)=0\) for \(p+q\geq n+1\), \(k\geq 1\) and \(\ell \geq n-p+r-1.\) The proof rests on a generalization of the Borel-Le Potier spectral sequence and the Kodaira-Akizuki-Nakano vanishing theorem for line bundles. Moreover it is shown that there is a canonical homomorphism \(H^{p,q}(X,\wedge^ 2E\otimes L)\to H^{p+1,q+1}(X,S^ 2E\otimes L)\) which is bijective under some additional hypotheses. Using this the author gives a counterexample to a conjecture of \textit{J. A. Sommese} in Math. Ann. 233, 229-256 (1978; Zbl 0381.14007).
scientific article; zbMATH DE number 4055754
scientific article