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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 PU(2)-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 ofPU(2)-instantons and the Kobayashi-Hitchin correspondence on surfaces.

Donaldson theory on non-K\"ahlerian surfaces and class $VII$ surfaces with $b_2=1$