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Note for a conference On manifolds whose tangent bundle contains
 

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), 593-606
J. Wahl Building up on Mori's work proved the theorem for r = 1
A cohomological characterization of P n , Inv. Math. 72 (1983), 315-322
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.

  

Source: Andreatta, Marco - Dipartimento di Matematica, UniversitÓ di Trento

 

Collections: Mathematics