Examples of Fano manifolds with non‐pseudoeffective tangent bundle

Abstract

Let X $X$ be a Fano manifold. While the properties of the anticanonical divisor − K X $-K_X$ and its multiples have been studied by many authors, the positivity of the tangent bundle T X $T_X$ is much more elusive. We give a complete characterisation of the pseudoeffectivity of T X $T_X$ for del Pezzo surfaces, hypersurfaces in the projective space and del Pezzo threefolds.