# On a result of Kawamata on ampleness and nonexistence of rational curves

Recently I read the following results.

(1) Zheng, F.. Kodaira dimensions and hyperbolicity of nonpositively curved compact K\"ahler manifolds. Comment. Math. Helv. 77 (2002), no. 2, 221-234.

(2) Jahnke, P.; Peternell, T.; Radloff, I.. Some recent developments in the classification theory of higher dimensional manifolds. Global aspects of complex geometry, 311-357, Springer, Berlin, 2006.

Theorem [Zheng Thm 2 p232] Let $M^2$ be a compact K\"ahler surface of nonpositive sectional curvature. If it is of general type, then it is Kobayashi hyperbolic.

Theorem [Jahnke-Peternell-Radloff Cor 6.10 p334] Let $X_n$ be a projective manifold whose universal cover is Stein (or has no positive-dimensional subvariety). Then either $K_X$ is ample or $\chi(\mathcal{O}_X) = 0$, $K_X$ is nef and $K_X^n=0$.

It seems that both results rely on the following step:

Proposition: Any minimal surface or variety $X_n$ of general type ($K_X$ big and nef), Assume $X_n$ is projective (no need when $n=2$), then $K_X$ is ample if $X_n$ admits no rational curve.

Both two papers refer this as a 1991 or 1992 result of Kawamata, however, I failed to find a precise statement. Can anyone give a precise statement of Kawamata's result or point out a reference?

So, the point is, if $K_X$ is nef and big, then by the basepoint-free theorem some multiple of it is basepoint-free and hence defines a morphism. It is easy to see that the image of that morphism is the canonical model of $X$ and hence has an ample canonical class.
To finish the argument one notes that if the original $X$ does not contain rational curves, then the above morphism must be an isomorphism and hence $K_X$ is ample.
• The Kahler case is in fact the same (in all dimensions) since $K_X$ big implies that $X$ is Moishezon, and since it is also Kahler, it is therefore projective by a result of Moishezon. So if $X$ is Kahler and $K_X$ is big and nef, then it is ample if $X$ contains no rational curve. – YangMills May 29 '18 at 9:04