We introduce orbifold Euler numbers for normal surfaces with boundary $\mathbb{Q}$-divisors. These numbers behave multiplicatively under finite maps and in the log canonical case we prove that they satisfy the Bogomolov–Miyaoka–Yau type inequality. Existence of such a generalization was earlier conjectured by G. Megyesi [Proc. London Math. Soc. (3) 78 (1999) 241–282]. Most of the paper is devoted to properties of local orbifold Euler numbers and to their computation.

As a first application we show that our results imply a generalized version 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 necessary condition for the logarithmic comparison theorem which recovers an 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 boundedness of rational curves in some surfaces of general type (conjectured by G. Tian [Springer Lecture Notes in Mathematics 1646 (1996) 143–185)]. Finally, we give some applications to singularities of plane curves; for example, we improve F. Hirzebruch's bound on the maximal number of cusps of a plane curve.

2000 Mathematical Subject Classification: 14J17, 14J29, 14C17.