  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 $X$.
