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Let be a normal projective variety, and let be an ample Cartier divisor on . Suppose that is not the projective space. We prove that the twisted cotangent sheaf is generically nef with respect to the polarisation . As an application we prove a Kobayashi-Ochiai theorem for foliations: if is a foliation such that , then is at most the rank of .
Let be a uniruled projective manifold and let be a general point. The main result of [2] says that if the -degrees (i.e., the degrees with respect to the anti-canonical bundle of ) of all rational curves through are at least , then is a projective space. In this paper, we study the structure of when the -degrees of all rational curves through are at least .
Our study uses the projective variety , called the VMRT at , defined as the union of tangent directions to the rational curves...
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