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## On negativity of total k-jet curvature and ampleness of the canonical bundle

A celebrated conjecture of Kobayashi and Lang says that the canonical line bundle K_X of a Kobayashi hyperbolic compact complex manifold X is ample. In this note we prove that K_X is ample if X is projective and satisfies a stronger condition of nondegenerate negative total k-jet curvature. We use positivity of direct image sheaves and decomposition of jets in order to produce pluridifferentials on X.