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Complex Algebraic Varieties pp 87–104Cite as

Singular hermitian metrics on positive line bundles

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Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1507))

Abstract

The notion of a singular hermitian metric on a holomorphic line bundle is introduced as a tool for the study of various algebraic questions. One of the main interests of such metrics is the corresponding L2 vanishing theorem for cohomology, which gives a useful criterion for the existence of sections. In this context, numerically effective line bundles and line bundles with maximum Kodaira dimension are characterized by means of positivity properties of the curvature in the sense of currents. The coefficients of isolated logarithmic poles of a plurisubharmonic singular metric are shown to have a simple interpretation in terms of the constant ε of Seshadri’s ampleness criterion. Finally, we use singular metrics and approximations of the curvature current to prove a new asymptotic estimate for the dimension of cohomology groups with values in high multiples O(kL) of a line bundle L with maximum Kodaira dimension.

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Authors and Affiliations

  1. Institut Fourier, BP 74, Laboratoire associé au C.N.R.S. no 188, Université de Grenoble I, F-38402, Saint-Martin d’Hères

    Jean-Pierre Damailly

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  1. Jean-Pierre Damailly
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Klaus Hulek Thomas Peternell Michael Schneider Frank-Olaf Schreyer

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© 1992 Springer-Verlag

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Damailly, JP. (1992). Singular hermitian metrics on positive line bundles. In: Hulek, K., Peternell, T., Schneider, M., Schreyer, FO. (eds) Complex Algebraic Varieties. Lecture Notes in Mathematics, vol 1507. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0094512

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  • DOI: https://doi.org/10.1007/BFb0094512

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-55235-2

  • Online ISBN: 978-3-540-46786-1

  • eBook Packages: Springer Book Archive

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