On the stability of the tangent bundle of a hypersurface in a Fano variety (Q2494643)

Let \(M\) be a complex Fano manifold of index \(i\) with Picard number equal to one. Let \(H\) be the ample generator of the Picard group and let \(Z\) be a smooth hypersurface that is a member of the linear system \(| dH| \). If the degree \(d\) is at least \(i\), the manifold \(Z\) has ample or trivial canonical bundle and it is well-known that the tangent bundle of \(Z\) is polystable. If the degree \(d\) is strictly smaller than \(i\), the manifold \(Z\) is Fano and in general its tangent bundle is not stable. The main result of this paper shows that if the tangent bundle of \(M\) is semistable, the restriction map \(\text{Pic}(M) \rightarrow \text{Pic}(Z)\) is an isomorphism, and the degree \(d\) is at least \(0.5 i\), the tangent bundle of \(Z\) is stable (the bound in the paper is slightly better and depends on the dimension of \(M\)). The main ingredient is a vanishing result for the top cohomology of twisted differential forms on \(Z\) which follows from the Akizuki-Nakano vanishing theorem and some standard exact sequences.