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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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