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Some remarks on vanishing theorems for holomorphic vector bundles. (English) Zbl 0543.32013
The main result of this paper is the following precise vanishing theorem: Let X be a projective manifold of dimension n and let E and F be vector bundles of rank r, resp. of rank 1 on X. Suppose that \({\mathcal O}_{{\mathbb{P}}(E)}(1)\) and \(A=(\det E)^{-1}\otimes K_ X^{-1}\otimes F\) are numerically semi-positive line bundles. Assume moreover that either \(c_ 1(A)^ n>0\) or that \(\tilde c_ n(E)>0.\) Then \(H^ q(X,S^{\mu}(E)\otimes F)=0\) for \(q>0,\quad \mu \geq 0.\) Here \(\tilde c{}_ n(E)\) is the component in \(H^{2n}(X,{\mathbb{Z}})\) of \(c(E^*)^{- 1}\). - This generalizes results of Griffiths. Similar improvements can be given for the vanishing theorems of Le Potier and Faltings.

32L20 Vanishing theorems
32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
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[1] Akizuki, Y., Nakano, S.: Note on Kodaira-Spencer’s proof of Lefschetz theorems. Proc. Japan Acad.30, 266-272 (1954) · Zbl 0059.14701 · doi:10.3792/pja/1195526105
[2] Barth, W.: Submanifolds of low codimension in projective space. Proceedings of the International Congress of Mathematicians (Vancouver 1974), pp. 409-413
[3] Demailly, J.P., Skoda, H.: Relations entre les notions de positivités de P.A. Griffiths et de S. Nakano pour les fibrés vectoriels. Séminaire P. Lelong, H. Skoda, 1978/79. Lecture Notes in Math.822, pp. 304-309. Berlin-Heidelberg-New York: Springer 1980
[4] Faltings, G.: Verschwindungssätze und Untermannigfaltigkeiten kleiner Kodimension des komplex-projektiven Raumes. J. Reine Angew. Math.326, 136-151 (1981) · Zbl 0452.14004 · doi:10.1515/crll.1981.326.136
[5] Fujita, T.: On Kähler fiber spaces over curves. J. Math. Soc. Japan30, 779-794 (1978) · Zbl 0393.14006 · doi:10.2969/jmsj/03040779
[6] Gieseker, D.:p-ample bundles and their Chern classes. Nagoya Math. J.,43, 91-116 (1971) · Zbl 0221.14010
[7] Grauert, H.: Über Modifikationen und exzeptionelle analytische Mengen. Math. Ann.146, 331-368 (1962) · Zbl 0173.33004 · doi:10.1007/BF01441136
[8] Griffiths, P.A.: Hermitian differential geometry, Chern classes, and positive vector bundles. In: Global Analysis, papers in honor of K. Kodaira pp. 185-251. Princeton: Princeton University Press 1969 · Zbl 0201.24001
[9] Hartshorne, R.: Ample vector bundles. Inst. Hautes Études Sci. Publ. Math.29, 63-94 (1966) · Zbl 0173.49003
[10] Hartshorne, R.: Ample subvarieties of algebraic varieties. Lecture Notes in Math.156. Berlin-Heidelberg-New York: Springer 1970 · Zbl 0208.48901
[11] Kawamata, Y.: A generalization of Kodaira-Ramanujam’s vanishing theorem. Math. Ann.261, 43-46 (1982) · Zbl 0488.14003 · doi:10.1007/BF01456407
[12] Kleiman, S.L.: Towards a numerical theory of ampleness. Ann. of Math.84, 293-344 (1966) · Zbl 0146.17001 · doi:10.2307/1970447
[13] Kodaira, K.: On a differential geometric method in the theory of analytic stacks. Proc. Nat. Acad. Sci. U.S.A.39, 1268-1273 (1953) · Zbl 0053.11701 · doi:10.1073/pnas.39.12.1268
[14] Nakano, S.: On complex analytic vector bundles. J. Math. Soc. Japan7, 1-12 (1955) · Zbl 0068.34403 · doi:10.2969/jmsj/00710001
[15] Peternell, Th.: On strongly pseudo-convex Kähler manifolds. Invent. Math.70, 157-168 (1982) · Zbl 0505.32023 · doi:10.1007/BF01390726
[16] le Potier, J.: Annulation de la cohomologie à valeurs dans un fibré vectoriel holomorphe positif de rang quelconque. Math. Ann.218, 35-53 (1975) · Zbl 0313.32037 · doi:10.1007/BF01350066
[17] Ramanujam, C.P.: Remarks on the Kodaira vanishing theorem. J. Indian Math. Soc.36, 41-51 (1972) · Zbl 0276.32018
[18] Ramanujam, C.P.: Supplement to the article ?Remarks on the Kodaira vanishing theorem?. J. Indian Math. Soc.38, 121-124 (1974) · Zbl 0368.14005
[19] Schmid, W.: Homogeneous complex manifolds and representations of semi-simple Lie groups. Proc. Nat. Acad. Sci. U.S.A.59, 56-59 (1968) · Zbl 0164.15803 · doi:10.1073/pnas.59.1.56
[20] Schneider, M.: Ein emfacher Beweis des Verschwindungssatzes für positive holomorphe Vektorraumbündel. Manuscripta Math.11, 95-101 (1974) · Zbl 0275.32014 · doi:10.1007/BF01189093
[21] Viehweg, E.: Vanishing theorems J. Reine Angew. Math.335, 1-8 (1982) · Zbl 0485.32019 · doi:10.1515/crll.1982.335.1
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