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

TL;DR

This paper proves that for certain K"ahler surfaces, key algebraic invariants like the canonical class are actually determined by the underlying smooth structure, confirming the Van de Ven Conjecture using Seiberg Witten invariants.

Contribution

It provides a self-contained proof linking Seiberg Witten invariants to the diffeomorphism invariance of canonical classes and elliptic surface properties, advancing understanding of differentiable structures in complex geometry.

Findings

Canonical class is a diffeomorphism invariant up to sign for certain K"ahler surfaces.

The Kodaira dimension is determined by the underlying differentiable manifold.

Elliptic fibration multiplicities are also diffeomorphism invariants.

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.