The Canonical Class and the C∞ Properties of Kähler Surfaces

摘要

We give a self contained proof thatfor 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 underlyingdifferentiable manifold.We then reprove that the multiplicities of the ellipticfibration are determined by the underlying oriented manifold, and thatthe plurigenera of a surface are oriented diffeomorphism invariants.We also compute the Seiberg Witten invariants of all Kählersurfaces of non-negative Kodaira dimension.The proof uses a set up of Seiberg Witten theory that replaces genericmetrics by the construction of a localised Euler class of an infinitedimensional bundle with a Fredholm section. This makes thetechniques of excess intersection available in gauge theory