 
Summary: Note for a conference
On manifolds whose tangent bundle contains
an ample subbundle
by Marco Andreatta
joint work with Jaros law A. Wisniewski which will be published on
Inventiones Mathematicae
MSC numb.: Prim.:14J40, 14J60, 14E30, 14M20
Let X be a complex projective manifold of dimension n and let E be a vector bundle of
rank r which is a subsheaf of the tangent sheaf TX, E ,! TX.
Theorem. If E is ample then X = P n and E = O(1) r or E = TP n .
History.
R.Hartshorne conjectured that P n is the only manifold whose tangent bundle is ample.
S. Mori proved the conjecture (i.e. the theorem for r = n) in a celebrated paper, which
contained an amazing proof of the existence of rational curves on Fano manifolds.
Projective manifolds with ample tangent bundles, Ann. Math. 110 (1979), 593606
J. Wahl Building up on Mori's work proved the theorem for r = 1
A cohomological characterization of P n , Inv. Math. 72 (1983), 315322
F. Campana and T. Peternell proved the theorem for r = n 1; n 2 and conjecture
the general theorem.
Rational curves and ampleness properties of tangent bundle of algebraic varieties, Manuscr.
