  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.
