Donaldson theory on non-Kählerian surfaces and class $VII$ surfaces with $b 2=1$ - Mathematics Differential GeometryReport as inadecuate




Donaldson theory on non-Kählerian surfaces and class $VII$ surfaces with $b 2=1$ - Mathematics Differential Geometry - Download this document for free, or read online. Document in PDF available to download.

Abstract: We prove that any class $VII$ surface with $b 2=1$ has curves. This impliesthe -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 isisomorphic to one of the surfaces in the known list. The main idea of the proofis to show that a certain moduli space of PU2-instantons on a surface $X$with no curves if such a surface existed would contain a closed Riemannsurface $Y$ whose general points correspond to non-filtrable holomorphicbundles 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 thealgebraicity of $Y$ to obtain a contradiction. The proof uses essentiallytechniques from Donaldson theory: compactness theorems for moduli spaces ofPU2-instantons and the Kobayashi-Hitchin correspondence on surfaces.



Author: Andrei Teleman

Source: https://arxiv.org/







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