## On manifolds whose tangent bundle contains an ample subbundle.(English)Zbl 1081.14060

In 1979 S. Mori proved a conjecture of Hartshorne, which stated that the only projective manifold with ample tangent bundle was the projective space itself; that paper [Ann. Math. (2) 110, 593–606 (1979; Zbl 0423.14006)] contained many new ideas and technical tools concerning the geometry of rational curves on projective variety, and has become a cornerstone in the study of higher dimensional algebraic varieties. In this paper a generalization of Mori’s result is proposed: if the tangent bundle of a projective manifold $$X$$ contains an ample locally free subsheaf $$E$$ then the variety is the projective space and the sheaf is the tangent bundle itself or a direct sum of hyperplane line bundles. Partial results in this directions were obtained by J. M. Wahl [Invent. Math. 72, 315–322 (1983; Zbl 0544.14013)], who considered the case in which the rank of $$E$$ is one and by F. Campana and Th. Peternell [Manuscr. Math. 97, No.1, 59–74 (1998; Zbl 0932.14024)], who considered the cases in which the rank of $$E$$ is equal to $$\dim X$$, $$\dim X-1$$ or $$\dim X-2$$. The proof, in the spirit of Mori’s ideas, makes use of families of rational curves on $$X$$, in particular it is based on the existence on $$X$$ of an unsplit family $$V$$ of rational curves; roughly speaking $$V$$ is a family of rational curves whose deformations don’t split up into reducible cycles. To this family one can associate an equivalence relation on $$X$$: two points are equivalent if there exists a chain of rational curves in $$V$$ joining the two points; this relation is called rational connectedness with respect to $$V$$. An highly non trivial fact is that to this relation, as proved by F. Campana [Ann. Sci. Éc. Norm. Supér., IV. Sér. 25, No. 5, 539–545 (1992; Zbl 0783.14022)] and J. Kollar, Y. Miyaoka, S. Mori [J. Differ. Geom. 36, No. 3, 765–779 (1992; Zbl 0759.14032)], one can associate a proper fibration, defined on an open subset of $$X$$ such that fibers of this map are equivalence classes for the rcV relation. In the assumptions of the main theorem it is proved that the rcV fibration is constant, i.e. any pair of points in $$X$$ can be joined by a chain of curves parametrized by $$V$$ (we say that $$X$$ is rcV connected). Then the splitting of $$E$$ is studied, showing that either $$E$$ is decomposable, reducing to Wahl’s theorem, or $$E$$ is the tangent bundle of $$X$$, reducing to Mori’s theorem.

### MSC:

 14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli 14E30 Minimal model program (Mori theory, extremal rays) 14M20 Rational and unirational varieties

### Keywords:

rational curves; tangent bundle
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