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

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