On generic nefness of tangent sheaves

We show that the tangent bundle of a projective manifold with nef anticanonical class is generically nef. That is, its restriction to curve cut out by general sufficiently ample divisors vector bundle. This confirms conjecture Peternell. As consequence, second Chern such has non-negative intersections divisors. also investigate under which conditions these positivities are strict, and answer question Yau.