On the stability of the tangent bundle of Fano manifolds

Abstract

By using the classification of Fano 3-folds we prove: Let $X$ be Fano 3-fold. Assume that the tangent bundle $T_X$ of $X$ is not stable (i.e. semi-stable or unstable). Then $b_2\ge 2$ and a relative tangent sheaf $T_{X/Y}$ of a contraction $f:X⟶Y$ of an extremal face on $X$ is a destabilising subsheaf of $T_X$.

arXiv comment9 pages, LaTeX, to appear in Math. Ann