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

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

2007-04-20

arxiv.org/abs/0704.2638v1

Inv. math, Volume 162, Number 3, December 2005, 493-521

Mathematics

Differential Geometry

LaTeX, 29 pages

Scientific paper

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. 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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