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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?

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I suppose they might mean the basepoint-free theorem. If you definitely want a reference to Kawamata, I believe it's in his Annals paper, "The cone of curves...", but probably the standard reference is Kollár-Mori (1998).

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.

Since it is a canonical model, it has rational Gorenstein singularities which implies that the (non-trivial) fibers of this morphism contain rational curves. In fact, a lot more is true, these fibers are rationally chain connected, so not only contain rational curves, but are covered by them. In its currently known form this was originally conjectured by Shokurov and proved by Hacon-McKernan. Their result is from 2007, so newer than the ones you are reading. Then again, you only need existence of rational curves which can be derived from the (relative) cone theorem for the morphism. Behind all of these is the phenomenal result of Mori which is probably best known as Bend-and-Break. That shows how the failure of the positivity of the canonical class leads to the existence of rational curves.

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.

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  • $\begingroup$ Thanks a lot! Can we say that in dim 2 and in dim 3, we only need Kahler other than being projective, due to some new developments in dim 3 in abundance. Sorry for any mistakes due to my poor knowledge in AG. $\endgroup$ – Bo_Y Jan 5 '17 at 3:39
  • $\begingroup$ 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. $\endgroup$ – YangMills May 29 '18 at 9:04

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