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

By A. Teleman. 2005

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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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/c94678da78d4f8b1acebb6045029bc0b92304072

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Monopole Floer homology for rational homology 3-spheres (2008, background): “Moreover, the global spherical shell conjecture for minimal class VII surfaces with b2 > 0 (known for b2 = 1, see [21]) would imply that L is defined and zero for such surfaces with the canonical spin structure.”
A parabolic flow of pluriclosed metrics (2009, background): “Thus the terms in the above summand with k > 0 are nonnegative multiples of the volume form of ω, and the result follows.”
The universal Kobayashi-Hitchin correspondence on Hermitian manifolds (2004, methodology): “Furthermore, important applications of the classical Kobayashi-Hitchin correspondence to the classification of non-K¨ahlerian surfaces can be found in [Te1], [ Te4 ] [LT], [LYZ].”
Regularity results for pluriclosed flow (2010, methodology): “9 implies the classification of Class VII surfaces with b2 = 1, a theorem obtained by Teleman using gauge theory [29].”
The Chern–Ricci flow on complex surfaces (2012, background): “These surfaces are of great interest since, except for the case b2(M) = 1 [52], they are not yet completely classified.”