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.

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

References in corpus (4)

• Algebraic surfaces and Seiberg-Witten invariants
• $L^2$ vanishing theorems for positive line bundles and adjunction theory, Lecture Notes of a CIME course on "Transcendental Methods of Algebraic Geometry" (Cetraro, Italy, July 1994)
• On complex surfaces diffeomorphic to rational surfaces
• Les Invariants de Seiberg-Witten et la Conjecture de Van De Ven

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