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

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dc.creatorTeleman, Andrei
dc.date2007-04-20
dc.date.accessioned2026-07-07T07:57:33Z
dc.date.available2026-07-07T07:57:33Z
dc.descriptionWe 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. 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.
dc.descriptionLaTeX, 29 pages
dc.identifierhttps://arxiv.org/abs/0704.2638
dc.identifierhttp://arxiv.org/abs/0704.2638
dc.identifierInv. math, Volume 162, Number 3, December 2005, 493-521
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/127686
dc.subjectDifferential Geometry
dc.subjectAlgebraic Geometry
dc.subjectComplex Variables
dc.subjectGeometric Topology
dc.subject53C55; 53C07; 32Q57; 32G13
dc.titleDonaldson theory on non-Kählerian surfaces and class $VII$ surfaces with $b_2=1$
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