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Logarithmic Orbifold Euler Numbers of Surfaces with Applications
Author(s) -
Langer Adrian
Publication year - 2003
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611502013874
Subject(s) - mathematics , orbifold , euler characteristic , euler's formula , ball (mathematics) , gravitational singularity , type (biology) , pure mathematics , logarithm , generalization , combinatorics , mathematical analysis , ecology , biology
We introduce orbifold Euler numbers for normal surfaces with boundary 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.