A line bundle over a complex projective variety is called big and 1-ample if a large multiple of it is generated by global sections and a morphism induced by the evaluation of the spanning sections is generically finite and has at most 1-dimensional fibers. A vector bundle is called big and 1-ample if the relative hyperplane line bundle over its projectivisation is big and 1-ample.
The main theorem of the present paper asserts that any complex projective manifold of dimension 4 or more, whose tangent bundle is big and 1-ample, is equal either to a projective space or to a smooth quadric. Since big and 1-ample bundles are ‘almost’ ample, the present result is yet another extension of the celebrated Mori paper ‘Projective manifolds with ample tangent bundles’ (Ann. of Math. 110 (1979) 593–606).
The proof of the theorem applies results about contractions of complex symplectic manifolds and of manifolds whose tangent bundles are numerically effective. In the appendix we re-prove these results.
