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, Andrei

Annales de l'Institut Fourier, Tome 65 (2015), p. 101-136 / Harvested from Numdam

Résumé

Soit $\pi \colon M\rightarrow B$ un submersion propre entre variétés complexes, et soit $\mathcal{E}$ un fibré holomorphe sur $M$. Nous étudions et décrivons explicitement le sous-faisceau de torsion $\mbox {Tors}(R^1\pi _*(\mathcal{E}))$ de la première image directe $R^1\pi _*(\mathcal{E})$ en supposant que $R^0\pi _*(\mathcal{E})=0$. 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 $H^0(\mbox {Tors}(R^1\pi _*(\mathcal{E})))$ dans une situation concrète liée à notre programme pour démontrer l’existence des courbes sur les surfaces de la classe VII.

Abstract

Let $\pi \colon M\rightarrow B$ be a proper holomorphic submersion between complex manifolds and $\mathcal{E}$ a holomorphic bundle on $M$. We study and describe explicitly the torsion subsheaf $\mbox {Tors}(R^1\pi _*(\mathcal{E}))$ of the first direct image $R^1\pi _*(\mathcal{E})$ under the assumption $R^0\pi _*(\mathcal{E})=0$. 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 $H^0(\mbox {Tors}(R^1\pi _*(\mathcal{E})))$ in a concrete situation related to our program to prove existence of curves on class VII surfaces.

Détail

Publié le : 2015-01-01

Zbl 06496535

DOI : https://doi.org/10.5802/aif.2926
Classification:  32C35,  32G05,  32J15
Mots clés: Faisceaux cohérents, images directes supérieures, surfaces complexes, déformation verselle, sous-faisceau de torsion

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