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Vanishing theorems for tensor powers of an ample vector bundle. (English) Zbl 0647.14005
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 \(q\geq 1\) [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 \(p+q\geq n+r\) [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 J. A. Sommese in Math. Ann. 233, 229-256 (1978; Zbl 0381.14007).
Reviewer: H.Lange

MSC:
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
32L20 Vanishing theorems
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References:
[1] Akizuki, Y., Nakano, S.: Note on Kodaira-Spencer’s proof of Lefschetz theorems. Proc. Jap. Acad.30, 266-272 (1954) · Zbl 0059.14701
[2] Borel, A., Weil, A.: Représentations linéaires et espaces homogènes kählériens des groupes de Lie compacts. Séminaire Bourbaki (exposé no 100 par J.-P. Serre), (mai 1954)
[3] Bott, R.: Homogeneous vector bundles. Ann. Math.66, 203-248 (1957) · Zbl 0094.35701
[4] Demailly, J.P.: Théorèmes d’annulation pour la cohomologie des puissances tensorielles d’un fibré positif. C.R. Acad. Sci. Paris Sér. I Math.,305, (1987) (à paraître.) · Zbl 0627.32022
[5] Demailly, J.P.: Vanishing theorems for tensor powers of a positive vector bundle. (to appear in the Proceedings of the Conference on Geometry and Analysis on Manifolds held in Katata, Japan (August 1987), Lect. Notes Math., Springer, Berlin Heidelberg New York) · Zbl 0627.32022
[6] Demazure, B.: A very simple proof of Bott’s theorem. Invent. Math.33, 271-272 (1976) · Zbl 0383.14017
[7] Godement, R.: Théorie des faisceaux. Hermann, Paris, 1958 · Zbl 0080.16201
[8] Griffiths, P.A.: Hermitian differential geometry, Chern classes and positive vector bundles. Global Analysis, Papers in honor of K. Kodaira, Princeton Univ. Press, Princeton (1969), pp. 185-251
[9] Hartshorne, R.: Ample vector bundles. Publ. Math. I.H.E.S.29, 63-94 (1966) · Zbl 0173.49003
[10] Kraft, H.: Geometrische Methoden in der Invariantentheorie. Aspekte der Mathematik, Band D 1, Braunschweig, Vieweg Sohn, 1985 · Zbl 0669.14003
[11] Peternell, Th., Le Potier, J., Schneider, M.: Vanishing theorems, linear and quadratic normality. Invent. Math.87, 573-586 (1987) · Zbl 0618.14023
[12] Peternell, Th., Le Potier, J., Schneider, M.: Direct images of sheaves of differentials and the Atiyah class. Math. Z.196, 75-85 (1987) · Zbl 0662.14006
[13] Le Potier, J.: Annulation de la cohomologie à valeurs dans un fibré vectoriel holomorphe de rang quelconque. Math. Ann.218, 35-53 (1975) · Zbl 0313.32037
[14] Schneider, M.: Ein einfacher Beweis des Verschwindungssatzes für positive holomorphe Vektorraumbündel. Manuscr. Math.11, 95-101 (1974) · Zbl 0275.32014
[15] Sommese, A.J.: Submanifolds of abelian varieties. Math. Ann.233, 229-256 (1978) · Zbl 0381.14007
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