The canonical class and the $C^\infty$ properties of K\"ahler surfaces

Abstract

We give a self contained proof using Seiberg Witten invariants that for K\"ahler surfaces with non negative Kodaira dimension (including those with $p_g = 0$) the canonical class of the minimal model and the $(-1)$-curves, are oriented diffeomorphism invariants up to sign. This implies that the Kodaira dimension is determined by the underlying differentiable manifold (Van de Ven Conjecture). We use a set up that replaces generic metrics by the construction of a localised Euler class of an infinite dimensional bundle with a Fredholm section. This allows us to compute the Seiberg Witten invariants of all elliptic surfaces with excess intersection theory. 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.