On the stability of the tangent bundle of Fano manifolds

Details On the stability of the tangent bundle of Fano manifolds On the stability of the tangent bundle of Fano manifolds

1994-07-10

arxiv.org/abs/alg-geom/9407004v1

Mathematics

Algebraic Geometry

9 pages, LaTeX, to appear in Math. Ann

Scientific paper

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 $b_2\geq 2$ and a relative tangent sheaf $T_{X/Y}$ of a
contraction $f:X\longrightarrow Y$ of an extremal face on $X$ is a
destabilising subsheaf of $T_X$.

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