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

Let $X$ be a smooth projective variety defined over an algebraically closed field of positive characteristic $p$ whose tangent bundle is nef. We prove that admits morphism $X \to M$ such the fibers are Fano varieties with nef and $T_M$ numerically flat. also extremal contractions exist as morphisms. As application, we that, if Frobenius can lifted modulo $p^2$ , then admits, up to finite étale ...