Euler Characteristics, Fubini’s Theorem, and the Riemann-hurwitz Formula

We relate Fubini’s theorem for Euler characteristics to Riemann-Hurwtiz formulae, and reprove a classical result of Iversen. The techniques used include algebraic geometry, complex geometry, and model theory. Possible applications to the study of wild ramification in finite characteristic are discussed. Introduction The first section of the paper reviews the concept of an Euler characteristic for a first order structure in model theory. The discussion is purely algebraic for the benefit of readers unfamiliar with model theory, and various examples are given. Once an Euler characteristic is interpreted as an integral, it is natural to ask whether Fubini’s theorem holds; that is, whether the order of integration can be interchanged in a repeated integral. In the second section we consider finite morphisms between smooth curves over any algebraically closed field, and show that Fubini’s theorem is almost equivalent to the RiemannHurwitz formula. More precisely, in characteristic zero the two are equivalent and so Fubini’s theorem is satisfied, whereas in finite characteristic the possible presence of wild ramification implies that, for any Euler characteristic, interchanging the order of integration is not always permitted. The third section discusses a notion weaker than the full Fubini property (a so-called strong Euler characteristic [12] [13]), but which is sufficent for our applications. We show that over an algebraically closed field of characteristic zero, there is exactly one strong Euler characteristic (over the complex numbers, this is the usual topological Euler characteristic). We return to finite morphisms between algebraic varieties, this time considering surfaces. Again, Fubini’s theorem is related to a Riemann-Hurwitz formula, originally due to Iversen [10]. Our methods provide a new proof of his result. The paper finishes with a discussion of the geometric approach to ramification theory of local fields. Acknowledgments The conference ’Motivic Integration and its Interactions with Model Theory and NonArchimedean Geometry’ at the ICMS, Edinburgh, during May 2008 encouraged me to think about these ideas. Part of this text was written while visiting the IHES, and I am grateful for the excellent working environment which this provided. This visit would not have been possible without the generosity of the Cecil King Foundation and the London Mathematical Society, in the form of the Cecil King Travel Scholarship. I thank my supervisor I. Fesenko for his constant encouragement. 2000 Mathematics Subject Classification 14E22 (primary), 03C60, 14G22, 14C17 (secondary). The author is supported by an EPSRC Doctoral Training Grant at the University of Nottingham