Exemples de faisceaux cohérents sans résolution localement libre en dimension 3

Vuletescu, Victor

doi:10.1016/j.crma.2012.03.020

Géométrie algébrique/Géométrie analytique

Exemples de faisceaux cohérents sans résolution localement libre en dimension 3
[Examples of coherent sheaves with no resolution by locally free sheaves in dimension 3]

Presented by: Claire Voisin

Vuletescu, Victor ^1,²

¹ Universitatea Bucureşti, Facultatea de Matematică şi Informatică, St. Academiei 14, Bucharest, Roumanie
² “Simion Stoilow” Institute of Mathematics of the Romanian Academy, Romania

Comptes Rendus. Mathématique, Volume 350 (2012) no. 7-8, pp. 411-412.

Abstract
Résumé

In 1982, Schuster proved that for any compact complex surface X, every coherent sheaf $F$ on X has global resolutions

\dots \to E_{n} \to E_{n - 1} \to \dots \to E_{1} \to E_{0} \to F \to 0

such that the

E_{i}

ʼs are locally free (vector bundles). In 2002, Voisin proved that this is false for some Kähler compact complex manifolds of dimension ⩾3. In this Note, we give some new examples of non-Kähler compact complex manifolds of dimension 3 and coherent sheaves

F

on X having no global resolution by vector bundles. The proof that these sheaves do not admit a locally free resolution is very different from Voisinʼs argument.

En 1982, Schuster a prouvé que pour toutes les surfaces complexes compactes lisses, tous les faisceaux cohérents $F$ sur X admettent des résolutions

\dots \to E_{n} \to E_{n - 1} \to \dots \to E_{1} \to E_{0} \to F \to 0

avec les

E_{i}

ʼs localement libres (fibrés vectoriels). En 2002, Voisin a prouvé que ceci est faux pour certaines variétés kähleriennes de dimension ⩾3. Dans cette Note, on donne de nouveaux exemples de variétés complexes compactes lisses X de dimension 3 non-kähleriennes et de faisceaux cohérents

F

sur X nʼadmettant pas globalement une résolution localement libre. La preuve que ces faisceaux nʼadmettent pas de résolution localement libre est très différente des arguments de Voisin.

Received: 2012-03-21
Accepted: 2012-03-30
Published online: 2012-04-01

DOI: 10.1016/j.crma.2012.03.020

Author's affiliations:

Vuletescu, Victor ^{1, 2}

¹ Universitatea Bucureşti, Facultatea de Matematică şi Informatică, St. Academiei 14, Bucharest, Roumanie
² “Simion Stoilow” Institute of Mathematics of the Romanian Academy, Romania

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     title = {Exemples de faisceaux coh\'erents sans r\'esolution localement libre en dimension 3},
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Vuletescu, Victor. Exemples de faisceaux cohérents sans résolution localement libre en dimension 3. Comptes Rendus. Mathématique, Volume 350 (2012) no. 7-8, pp. 411-412. doi : 10.1016/j.crma.2012.03.020. https://www.numdam.org/articles/10.1016/j.crma.2012.03.020/

References
Cited by

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[2] Blanchard, A. Espaces fibrés kählériens compacts, C. R. Acad. Sci. Paris, Ser. I, Volume 238 (1954), pp. 2281-2283

[3] Brînzǎnescu, V. The Neron–Severi group for nonalgebraic surfaces I. Elliptic bundle case, Manuscripta Math., Volume 79 (1993) no. 2, pp. 187-195

[4] Höfer, T. Remarks on torus principal bundles, J. Math. Kyoto Univ. (JMKYAZ), Volume 33 (1993) no. 1, pp. 227-259

[5] Schuster, H.W. Locally free resolutions of coherent sheaves on surfaces, J. Reine Angew. Math., Volume 337 (1982), pp. 159-165

[6] Voisin, C. A counterexample to the Hodge conjecture extended to Kähler varieties, Int. Math. Res. Not., Volume 20 (2002), pp. 1057-1075

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