 
Summary: On manifolds whose tangent bundle contains
an ample subbundle
Marco Andreatta and Jaros law A. Wisniewski
Introduction.
Let X be a complex projective manifold of dimension n and let E be a vector bundle of
rank r, or equivalently a locally free OX sheaf of rank r. The bundle E is called ample if the
relative hyperplane line bundle OP(E) (1) over its projectivisation P(E) = P roj X (Sym(E))
is ample. We will assume that E is a subsheaf of the tangent sheaf TX, that is there exists
an injective morphism E ,! TX. In this paper we will prove the following:
Theorem. If E is an ample locally free subsheaf of TX then X = P n and E = O(1) r
or E = TP n .
The characterization of P n as the only manifold whose tangent bundle is ample was
conjectured by R.Hartshorne. The Hartshorne conjecture was proved by S. Mori in a
celebrated paper [Mo], which contained an amazing proof of the existence of rational
curves on Fano manifolds. Building up on Mori's work a version of the present theorem
was successively proved for r = 1 and r = n; n 1; n 2 by J. Wahl [Wa] and, respectively,
by F. Campana and T. Peternell [CP]. Moreover Campana and Peternell posed a question
with the above characterization of P n which generalizes previous results.
The proof of the main theorem will apply rational curves on X. Our notation is con
sistent with the book of J. Kollar ([Ko]). In particular Hom(P 1 ; X) denotes the scheme
