On the torsion of the first direct image of a locally free sheaf

Teleman, Andrei

Annales de l'Institut Fourier, Tome 65 (2015) no. 1, pp. 101-136.

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

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

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