Donaldson theory on non-Kählerian surfaces and class VII surfaces with b2=1

Abstract

We prove that any class VII surface with b2=1 has curves. This implies the “Global Spherical Shell conjecture” in the case b2=1: Any minimal class VII surface withb2=1 admits a global spherical shell, hence it is isomorphic to one of the surfaces in the known list. By the results in [LYZ], [Te1], which treat the case b2=0 and give complete proofs of Bogomolov’s theorem, one has a complete classification of all class VII-surfaces with b2∈{0,1}. The main idea of the proof is to show that a certain moduli space of PU(2)-instantons on a surface X with no curves (if such a surface existed) would contain a closed Riemann surface Y whose general points correspond to non-filtrable holomorphic bundles on X. Then we pass from a family of bundles on X parameterized by Y to a family of bundles on Y parameterized by X, and we use the algebraicity of Y to obtain a contradiction. The proof uses essentially techniques from Donaldson theory: compactness theorems for moduli spaces of PU(2)-instantons and the Kobayashi-Hitchin correspondence on surfaces.