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

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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.

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  1. LATP, CMI, Université de Provence, 39 Rue F. Joliot-Curie, 13453, Marseille Cedex 13, France

    Andrei Teleman

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  1. Andrei Teleman
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Correspondence to Andrei Teleman.

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Teleman, A. Donaldson theory on non-Kählerian surfaces and class VII surfaces with b2=1. Invent. math. 162, 493–521 (2005). https://doi.org/10.1007/s00222-005-0451-2

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  • DOI: https://doi.org/10.1007/s00222-005-0451-2

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