Strictly Nef Divisors
  Let $X$ be a projective manifold of dimension $n$ and $L$ a strictly nef line bundle on $X$. Then $K_X+tL$ is ample if $t > n+1$ in the following cases.   1.) $\text{dim} X = 3$ unless (possibly) $X$ is a Calabi-Yau with $c_2 \cdot L=0$;   2.) $\kappa(X) \ge n-2$;   3.) $\text{dim} \alpha(X) \ge n-2$, with $\alpha: X \to A$ the Albanese map. 
Fr\'ed\'eric Campana, Jungkai A. Chen, Thomas Peternell