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Miyaoka’s theorems on the generic seminegativity of $$T_ X$$ and on the Kodaira dimension of minimal regular threefolds. (English) Zbl 0809.14034
Kollár, János (ed.), Flips and abundance for algebraic threefolds. A summer seminar at the University of Utah, Salt Lake City, 1991. Paris: Société Mathématique de France, Astérisque. 211, 103-114 (1992).
This paper is part of a seminar devoted to the proof of the so-called “abundance conjecture” for threefolds.
The author gives here a new proof of some results due to Y. Miyaoka, in particular:
(1) if $$X$$ is a normal not uniruled complex variety, then $$\Omega^ 1_ X$$ is generically semipositive,
(2) if $$X$$ is a minimal regular threefold, then its Kodaira dimension $$k(X) \geq 0$$.
The main argument of the proof is the existence of some special rational curves on a variety $$X$$ whose tangent bundle is not generically seminegative.
For the entire collection see [Zbl 0782.00075].

##### MSC:
 14J30 $$3$$-folds 14C20 Divisors, linear systems, invertible sheaves
##### Keywords:
abundance conjecture; threefolds; Kodaira dimension