We study various “generic'' nefness and ampleness notions for holomorphic vector bundles on a projective manifold. We apply this in particular to the tangent bundle and investigate the relation to the geometry of the manifold.
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Thomas Peternell, Varieties with generically nef tangent bundles. J. Eur. Math. Soc. 14 (2012), no. 2, pp. 571–603