On the torsion of the first direct image of a locally free sheaf
Annales de l'Institut Fourier, Volume 65 (2015) no. 1, p. 101-136

Let π:MB be a proper holomorphic submersion between complex manifolds and a holomorphic bundle on M. We study and describe explicitly the torsion subsheaf Tors(R 1 π * ()) of the first direct image R 1 π * () under the assumption R 0 π * ()=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 * ())) in a concrete situation related to our program to prove existence of curves on class VII surfaces.

Soit π:MB un submersion propre entre variétés complexes, et soit un fibré holomorphe sur M. Nous étudions et décrivons explicitement le sous-faisceau de torsion Tors(R 1 π * ()) de la première image directe R 1 π * () en supposant que R 0 π * ()=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 * ())) dans une situation concrète liée à notre programme pour démontrer l’existence des courbes sur les surfaces de la classe VII.

DOI : https://doi.org/10.5802/aif.2926
Classification:  32C35,  32G05,  32J15
Keywords: coherent sheaves, higher direct images, complex surfaces, versal deformation, torsion subsheaf
@article{AIF_2015__65_1_101_0,
     author = {Teleman, Andrei},
     title = {On the torsion of the first direct image of a locally free sheaf},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {65},
     number = {1},
     year = {2015},
     pages = {101-136},
     doi = {10.5802/aif.2926},
     zbl = {06496535},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2015__65_1_101_0}
}
Teleman, Andrei. On the torsion of the first direct image of a locally free sheaf. Annales de l'Institut Fourier, Volume 65 (2015) no. 1, pp. 101-136. doi : 10.5802/aif.2926. http://www.numdam.org/item/AIF_2015__65_1_101_0/

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