Quantitative hyperbolicity for complex manifolds via numerical invariants

We introduce numerical invariants called hyperbolic indices, which measure the hyperbolicity of compact Kähler manifolds using directed positive closed currents. We prove that if a manifold $X$ has positive hyperbolic indices, then $X$ is Kobayashi hyperbolic; and if $X$ satisfies Demailly’s condition of negative jet curvature, then it has positive hyperbolic indices. In particular, by combining the method of jet differentials and density currents, we can prove that for a general hypersurface $X_d$ of degree $d$ in ${\mathbb{P}}^{n+1}$, the hyperbolic indices of $X_d$ grows to $\mathrm{\infty }$ with at least linear growth in $d$. Finally, we discuss an analytic approach to the Kobayashi conjecture.