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

Chen, Xi; Kerr, Matt; Lewis, James D.

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
[Le faisceau des fonctions méromorphes non nulles sur une variété algébrique projective n'est pas acyclique]

Présenté par : Jean-Pierre Demailly

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

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

Abstract (VO)
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.

Reçu le : 2010-01-27
Accepté le : 2010-02-02
Publié le : 2010-02-20

DOI : 10.1016/j.crma.2010.02.008

Affiliations des auteurs :

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

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     author = {Chen, Xi and Kerr, Matt and Lewis, James D.},
     title = {The sheaf of nonvanishing meromorphic functions in the projective algebraic case is not acyclic},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {291--293},
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Chen, Xi; Kerr, Matt; Lewis, James D. The sheaf of nonvanishing meromorphic functions in the projective algebraic case is not acyclic. Comptes Rendus. Mathématique, Tome 348 (2010) no. 5-6, pp. 291-293. doi: 10.1016/j.crma.2010.02.008

Bibliographie
Cité par

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

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

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

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