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THE BOGOMOLOV–MIYAOKA–YAU INEQUALITY FOR LOG CANONICAL SURFACES

Published online by Cambridge University Press:  30 October 2001

ADRIAN LANGER
Affiliation:
Instytut Matematyki UW, ul. Banacha 2, 02-097 Warszawa, Poland; alan@mimuw.edu.pl Current address: Mathematics Institute, Warwick University, Coventry CV4 7AL; langer@maths.warwick.ac.uk
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Abstract

Let X be a smooth projective surface of non-negative Kodaira dimension. Bogomolov [1, Theorem 5] proved that c21 [les ] 4c2. This was improved to c21 [les ] 3c2 by Miyaoka [12, Theorem 4] and Yau [19, Theorem 4]. Equality c21 [les ] 3c2 is attained, for example, if the universal cover of X is a ball (if κ(X) = 2 then this is the only possibility). Further generalizations of inequalities for Chern classes for some singular surfaces with (fractional) boundary were obtained by Sakai [16, Theorem 7.6], Miyaoka [13, Theorem 1.1], Kobayashi [6, Theorem 2; 7, Theorem 12], Wahl [18, Main Theorem] and Megyesi [10, Theorem 10.14; 11, Theorem 0.1].

In [8] we introduced Chern classes of reflexive sheaves, using Wahl's local Chern classes of vector bundles on resolutions of surface singularities. Here we apply them to obtain the following generalization of the Bogomolov–Miyaoka–Yau inequality.

Type
Research Article
Copyright
The London Mathematical Society 2001

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