On the Kodaira dimension of minimal threefolds (Q579358)

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 \textit{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).