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The Chern classes and Kodaira dimension of a minimal variety. (English) Zbl 0648.14006
Algebraic geometry, Proc. Symp., Sendai/Jap. 1985, Adv. Stud. Pure Math. 10, 449-476 (1987).
[For the entire collection see Zbl 0628.00007.]
Let X be a n-dimensional normal projective $${\mathbb{Q}}$$-Gorenstein variety (over a field of characteristic $$0)$$ with singular locus of codimension $$\geq 3$$ and with numerically effective canonical divisor; let $$f: Y\to X$$ be any resolution of the singularities.
This paper is devoted to prove the following inequality of Bogomolov type for the Chern classes of $$Y: (3c_ 2(Y)-c^ 2_ 1(Y))(f^*H_ 1...f^*H_{n-2})\geq 0$$ for arbitrary ample divisors $$H_ 1,...,H_{n-2}$$ on X.
To this aim the author at first studies in detail the notion of $${\mathfrak H}$$-semistable bundle for a given n-1-tuple $${\mathfrak H}=(h_ 1,...,h_{n-1})$$ of numerically effective $${\mathbb{Q}}$$-Cartier divisors; in particular he extends to the n-dimensional case the Bogomolov-Gieseker inequality for semistable bundles on a surface. Then he applies these results to the case of a normal projective not uniruled variety to prove the generic $$(H_ 1,...,H_{n-2})$$-semipositivity of the cotangent bundle of Y. The combined use of these results gives the required inequality and, in the case $$n=3$$, the non-negativity of the Kodaira dimension for some minimal 3-folds, in particular for a Gorenstein normal projective 3-fold with only canonical singularities and numerically effective canonical divisor.
Reviewer: L.Picco Botta

##### MSC:
 14E30 Minimal model program (Mori theory, extremal rays) 14C20 Divisors, linear systems, invertible sheaves 14J30 $$3$$-folds 57R20 Characteristic classes and numbers in differential topology