Logarithmic Orbifold Euler Numbers of Surfaces with Applications

We introduce orbifold Euler numbers for normal surfaces withboundary Q-divisors. These numbers behave multiplicatively underfinite maps and in the log canonical case we prove that theysatisfy the Bogomolov–Miyaoka–Yau type inequality.Existence of such a generalization was earlier conjectured byG. Megyesi Proc. London Math. Soc. (3) 78 (1999) 241–282].Most of the paper is devoted to properties of local orbifoldEuler numbers and to their computation. As a first application we show that our results imply a generalizedversion of R. Holzapfel's ‘proportionality theorem’Ball and surface arithmetics, Aspects of Mathematics E29 (Vieweg,Braunschweig, 1998)]. Then we show a simple proof of a necessarycondition for the logarithmic comparison theorem which recoversan earlier result by F. Calderón-Moreno, F. Castro-Jiménez,D. Mond and L. Narváez-Macarro Comment. Math. Helv.77 (2002) 24–38]. Then we prove effective versions of Bogomolov's result on boundednessof rational curves in some surfaces of general type (conjecturedby G. Tian Springer Lecture Notes in Mathematics 1646 (1996)143–185)]. Finally, we give some applications to singularitiesof plane curves; for example, we improve F. Hirzebruch's boundon the maximal number of cusps of a plane curve. 2000 MathematicalSubject Classification: 14J17, 14J29, 14C17.