[Fibrés vectoriels holomorphes sur les surfaces non algébriques]
Le problème de l'existence des structures holomorphes sur les fibrés vectoriels au-dessus des surfaces non algébriques est en général encore ouvert. Nous résolvons ce problème pour les fibrés de rang 2 sur les surfaces K3 et pour les fibrés de rangs arbitraires sur toutes les surfaces connues de la classe VII. Nos méthodes, qui s'appuient sur la théorie de Donaldson et sur la théorie des déformations, peuvent être utilisées pour résoudre le problème de l'existence des fibrés vectoriels holomorphes sur d'autres classes de surfaces non algébriques.
The existence problem for holomorphic structures on vector bundles over non-algebraic surfaces is, in general, still open. We solve this problem in the case of rank 2 vector bundles over K3 surfaces and in the case of vector bundles of arbitrary rank over all known surfaces of class VII. Our methods, which are based on Donaldson theory and deformation theory, can be used to solve the existence problem of holomorphic vector bundles on further classes of non-algebraic surfaces.
Accepté le :
Publié le :
@article{CRMATH_2002__334_5_383_0,
author = {Teleman, Andrei and Toma, Matei},
title = {Holomorphic vector bundles on non-algebraic surfaces},
journal = {Comptes Rendus. Math\'ematique},
pages = {383--388},
publisher = {Elsevier},
volume = {334},
number = {5},
year = {2002},
doi = {10.1016/S1631-073X(02)02278-1},
language = {en},
url = {http://archive.numdam.org/articles/10.1016/S1631-073X(02)02278-1/}
}
TY - JOUR AU - Teleman, Andrei AU - Toma, Matei TI - Holomorphic vector bundles on non-algebraic surfaces JO - Comptes Rendus. Mathématique PY - 2002 SP - 383 EP - 388 VL - 334 IS - 5 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/S1631-073X(02)02278-1/ DO - 10.1016/S1631-073X(02)02278-1 LA - en ID - CRMATH_2002__334_5_383_0 ER -
%0 Journal Article %A Teleman, Andrei %A Toma, Matei %T Holomorphic vector bundles on non-algebraic surfaces %J Comptes Rendus. Mathématique %D 2002 %P 383-388 %V 334 %N 5 %I Elsevier %U http://archive.numdam.org/articles/10.1016/S1631-073X(02)02278-1/ %R 10.1016/S1631-073X(02)02278-1 %G en %F CRMATH_2002__334_5_383_0
Teleman, Andrei; Toma, Matei. Holomorphic vector bundles on non-algebraic surfaces. Comptes Rendus. Mathématique, Tome 334 (2002) no. 5, pp. 383-388. doi : 10.1016/S1631-073X(02)02278-1. http://archive.numdam.org/articles/10.1016/S1631-073X(02)02278-1/
[1] M. Aprodu, V. Brı̂nzănescu, M. Toma, Holomorphic vector bundles on primary Kodaira surfaces, Math. Z. (to appear)
[2] Sur l'existence des fibrés vectoriels holomorphes sur les surfaces non-algébriques, J. Reine Angew. Math., Volume 378 (1987), pp. 1-31
[3] The Geometry of Four-Manifolds, Oxford University Press, 1990
[4] Smooth 4-Manifolds and Complex Surfaces, Springer-Verlag, 1994
[5] On surfaces of class VII0, Invent. Math., Volume 24 (1974), pp. 269-310
[6] Embedded surfaces and the structure of Donaldson's polynomial invariants, J. Differential Geom., Volume 41 (1995), pp. 573-734
[7] The Kobayashi–Hitchin Correspondence, World Scientific, 1995
[8] On surfaces of class VII0 with curves, II, Tôhoku Math. J., Volume 42 (1990), pp. 475-516
[9] Donaldson polynomials for K3 surfaces, J. Differential Geom., Volume 35 (1992), pp. 415-427
[10] Moduli of Kähler manifolds equipped with Hermite–Einstein vector bundles, Rev. Roumaine Math. Pures Appl., Volume 38 (1993), pp. 703-719
[11] Projectively flat surfaces and Bogomolov's theorem on class VII0 surfaces, Internat. J. Math., Volume 5 (1994), pp. 253-264
[12] Une classe de fibrés vectoriels holomorphes sur les 2-tores complexes, C. R. Acad. Sci. Paris, Volume 311 (1990), pp. 257-258
[13] Stable bundles on non-algebraic surfaces giving rise to compact moduli spaces, C. R. Acad. Sci. Paris, Volume 323 (1996), pp. 501-505
[14] Stable bundles with small c2 over 2-dimensional complex tori, Math. Z., Volume 232 (1999), pp. 511-525
Cité par Sources :
