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

الأصل · EN

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(R¹π*(E)) of the first direct image R¹π*(E) under the assumption R⁰π*(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 H⁰(Tors(R¹π*(E))) in a concrete situation related to our program to prove the existence of curves on class VII surfaces.