The sheaf of nonvanishing meromorphic functions in the projective algebraic case is not acyclic

Xi Chen; Matt Kerr; James D. Lewis

doi:10.1016/j.crma.2010.02.008

Algebraic Geometry/Analytic Geometry

The sheaf of nonvanishing meromorphic functions in the projective algebraic case is not acyclic

Jean-Pierre Demailly

Xi Chen ¹; Matt Kerr ²; James D. Lewis ¹

¹632 Central Academic Building, University of Alberta, Edmonton, Alberta T6G 2G1, Canada
²104 Math. Sciences Bldg., Durham University, Durham, DH1 3LE, UK

Comptes Rendus. Mathématique, Volume 348 (2010) no. 5-6, pp. 291-293

Abstract (Original language)
Résumé

Let $X / C$ be a projective algebraic manifold, and $M_{X}^{*}$ be the sheaf of nonvanishing meromorphic functions on X in the analytic topology. We prove a number of nonvanishing results for $H^{•} (X, M_{X}^{*})$ . In particular, $M_{X}^{*}$ is acyclic iff $\dim X = 1$ .

Sur une variété algébrique projective lisse $X / C$ , soit $M_{X}^{*}$ le faisceau des germes de fonctions méromorphes non nulles pour la topologie analytique de X. Nous démontrons un certain nombre de résultats de non annulation pour la cohomologie $H^{•} (X, M_{X}^{*})$ . En particulier, le faisceau $M_{X}^{*}$ est acyclique si et seulement si X est de dimension 1.

Article information
Cite this article

Received: 2010-01-27
Accepted: 2010-02-02
Published online: 2010-02-20

DOI: 10.1016/j.crma.2010.02.008

Author's affiliations:

Xi Chen ¹ ; Matt Kerr ² ; James D. Lewis ¹

¹ 632 Central Academic Building, University of Alberta, Edmonton, Alberta T6G 2G1, Canada
² 104 Math. Sciences Bldg., Durham University, Durham, DH1 3LE, UK

Xi Chen; Matt Kerr; James D. Lewis. The sheaf of nonvanishing meromorphic functions in the projective algebraic case is not acyclic. Comptes Rendus. Mathématique, Volume 348 (2010) no. 5-6, pp. 291-293. doi: 10.1016/j.crma.2010.02.008

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References
Cited by

[1] E. Bombieri; D. Husemöller Classification and embeddings of surfaces, Proceedings of Symposia in Pure Mathematics, Volume 29 (1975), pp. 329-420

[2] J. Dieudonné Panorama des Mathématiques Pures, Gauthier–Villars, 1977

[3] J.D. Lewis A Survey of the Hodge Conjecture, CRM Monograph Series, vol. 10, AMS, Providence, 1999

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