## On the Kodaira dimension of minimal threefolds.(English)Zbl 0625.14023

We show that a complex threefold X has Kodaira dimension $$\geq 0$$ if X admits a minimal model. In view of a recent result of S. Mori [“Flip theorem and the existence of minimal models for 3-folds”, J. Am. Math. Soc. 1 (1988), to appear], our theorem amounts to the following characterization of threefolds of Kodaira dimension $$-\infty:$$ For a complex projective threefold X, $$\kappa (X)=-\infty$$ if and only if X is uniruled.
The proof is a combination of algebro-geometric results (the pseudo- effectivity of $$c_ 2$$ and the generic semi-positivity of $$\Omega ^ 1_ X)$$ and the differential geometric one (Donaldson’s characterization of flat vector bundles on an algebraic surface).

### MSC:

 14J30 $$3$$-folds 14E30 Minimal model program (Mori theory, extremal rays) 14J25 Special surfaces
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### References:

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