New York Journal of Mathematics
Volume 2 (1996) 103-146


Rogier Brussee

The Canonical Class and the C Properties of Kähler Surfaces

Published: December 31, 1996
Keywords: Surfaces, 4-manifolds, Seiberg Witten-theory, ∞-dimensional intersection theory
Subject: Primary: 14J, 57N13. Secondary: 58B, 57R20

We give a self contained proof that for Kähler surfaces with non-negative Kodaira dimension, the canonical class of the minimal model and the (-1)-curves are oriented diffeomorphism invariants up to sign. This includes the case pg = 0. It implies that the Kodaira dimension is determined by the underlying differentiable manifold. We then reprove that the multiplicities of the elliptic fibration are determined by the underlying oriented manifold, and that the plurigenera of a surface are oriented diffeomorphism invariants. We also compute the Seiberg Witten invariants of all Kähler surfaces of non-negative Kodaira dimension. The proof uses a set up of Seiberg Witten theory that replaces generic metrics by the construction of a localised Euler class of an infinite dimensional bundle with a Fredholm section. This makes the techniques of excess intersection available in gauge theory

Author information

Fakultät für Mathematik Universität Bielefeld, Postfach 100131, 33501 Bielefeld