no. 1
On the torsion of the first direct image of a locally free sheaf
[Sur la torsion de la première image directe d’un faisceau localement libre]
Teleman, Andrei1
1 Aix-Marseille Université CNRS, Centrale Marseille, I2M, UMR 7373 13453 Marseille (France)
Annales de l'Institut Fourier, Tome 65 (2015) no. 1, pp. 101-136.

Soit π:M→B un submersion propre entre variétés complexes, et soit ℰ un fibré holomorphe sur M. Nous étudions et décrivons explicitement le sous-faisceau de torsion Tors(R 1 π * (ℰ)) de la première image directe R 1 π * (ℰ) en supposant que R 0 π * (ℰ)=0. Nous discutons deux applications des résultats obtenus  : la première concerne le lieu des points où une famille génériquement verselle de surfaces complexes est non-verselle. La deuxième application est un résultat d’annulation pour H 0 (Tors(R 1 π * (ℰ))) dans une situation concrète liée à notre programme pour démontrer l’existence des courbes sur les surfaces de la classe VII.

Let π:M→B be a proper holomorphic submersion between complex manifolds and ℰ a holomorphic bundle on M. We study and describe explicitly the torsion subsheaf Tors(R 1 π * (ℰ)) of the first direct image R 1 π * (ℰ) under the assumption R 0 π * (ℰ)=0. We give two applications of our results. The first concerns the locus of points in the base of a generically versal family of complex surfaces where the family is non-versal. The second application is a vanishing result for H 0 (Tors(R 1 π * (ℰ))) in a concrete situation related to our program to prove existence of curves on class VII surfaces.

DOI : 10.5802/aif.2926
Classification : 32C35, 32G05, 32J15
Keywords: coherent sheaves, higher direct images, complex surfaces, versal deformation, torsion subsheaf
Mot clés : Faisceaux cohérents, images directes supérieures, surfaces complexes, déformation verselle, sous-faisceau de torsion

Teleman, Andrei 1

1 Aix-Marseille Université CNRS, Centrale Marseille, I2M, UMR 7373 13453 Marseille (France) 
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Teleman, Andrei. On the torsion of the first direct image of a locally free sheaf. Annales de l'Institut Fourier, Tome 65 (2015) no. 1, pp. 101-136. doi : 10.5802/aif.2926. https://aif.centre-mersenne.org/articles/10.5802/aif.2926/

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