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Article Dans Une Revue Inventiones Mathematicae Année : 2005

Donaldson theory on non-Kählerian surfaces and class VII surfaces with b_2=1

Andrei Teleman

Résumé

We prove that any class VII surface with b_2=1 has curves. This implies the "Global Spherical Shell conjecture" in the case b_2=1: Any minimal class VII surface with b_ 2=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 b_2=0 and give complete proofs of Bogomolov's theorem, one has a complete classification of all class VII-surfaces with b_2∈{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.

Dates et versions

hal-00881266 , version 1 (08-11-2013)

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Citer

Andrei Teleman. Donaldson theory on non-Kählerian surfaces and class VII surfaces with b_2=1. Inventiones Mathematicae, 2005, 162 (3), pp.493-521. ⟨10.1007/s00222-005-0451-2⟩. ⟨hal-00881266⟩
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