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Vanishing theorems and Kählerity for strongly pseudoconvex manifolds

Author: Vo Van Tan
Journal: Trans. Amer. Math. Soc. 261 (1980), 297-302
MSC: Primary 32L20; Secondary 32F30, 53C55
Correction: Trans. Amer. Math. Soc. 291 (1985), 379-380.
MathSciNet review: 576877
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Abstract: A precise vanishing theorem of Kodaira-Nakano type for strongly pseudoconvex manifolds and Nakano semipositive vector bundles is established. This result answers affirmatively a question posed by Grauert and Riemenschneider. However an analogous version of vanishing theorem of Akizuki-Nakano type for strongly pseudoconvex manifolds and Nakano semipositive line bundles does not hold in general. A counterexample for this fact is explicitly constructed. Furthermore we prove that any strongly pseudoconvex manifold with 1-dimensional exceptional subvariety is Kählerian; in particular any strongly pseudoconvex surface is Kählerian.

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Keywords: Strongly pseudoconvex manifolds, Nakano semipositive vector bundles, weakly negative line bundles, precise vanishing theorem, non-Kählerian strongly pseudoconvex manifolds, Kähler metric
Article copyright: © Copyright 1980 American Mathematical Society

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