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The Canonical Class and the $$C^ \infty$$ Properties of Kähler Surfaces. (English) Zbl 0881.53057
Summary: 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 $$p_g = 0$$. 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 setup 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.

##### MSC:
 53C55 Global differential geometry of Hermitian and Kählerian manifolds 57N13 Topology of the Euclidean $$4$$-space, $$4$$-manifolds (MSC2010) 58B05 Homotopy and topological questions for infinite-dimensional manifolds 57R20 Characteristic classes and numbers in differential topology
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