The canonical class and the $C^\infty$ properties of Kähler surfaces
Abstract
We give a self contained proof using Seiberg Witten invariants that for Kähler 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.
 Publication:

arXiv eprints
 Pub Date:
 March 1995
 arXiv:
 arXiv:alggeom/9503004
 Bibcode:
 1995alg.geom..3004B
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Differential Geometry;
 57N13 (primary) 47H11 (secondary)
 EPrint:
 38 pages. Hardcopy available upon request. Latex 2e with amsart v. 1.2 or AMSlaTeX version 1.1. reason for resubmission: Changed title, corrected serious error in the argument for $p_g = 0$, major technical improvements in the handling of the localised Euler class of infinite dimensional bundles, referred some analysis to the literature and made some general improvements in the exposition