|In 1979, S. Mori characterized projective spaces as the only
ample tangent bundles. Since then, characterizations of projective spaces in terms of
positivity properties of the tangent bundle have been subject of much investigation.
In this talk I will discuss a recent characterization of projective spaces and
hyperquadrics that had been conjectured by Beauville. Namely, if the p-th exterior power
of the tangent bundle of a complex manifold X contains the p-th power of an ample line
bundle, then X is either the projective space or the p-dimensional hyperquadric.
This is a joint work with Stéphane Druel and Sàndor Kovàcs.