The Chern classes and Kodaira dimension of a minimal variety. (English) Zbl 0648.14006

Miyaoka, Yoichi

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

Keywords:

Gorenstein variety; resolution of the singularities; Chern classes; Bogomolov-Gieseker inequality; minimal 3-folds; numerically effective canonical divisor

Citations:

Zbl 0628.00007