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Let $D$ denote the disk $|z|<1$ in the complex plane and $U=D\0$(punctured disk). Define a holomorphic connection $\nabla$ on $\mathscr{O}_U$ by $\nabla(1)=\exp{(-1/z)}$. Does this extend to a logarithmic connection on $\mathscr{O}_D$, i.e. does this extend to $\nabla_D:\mathscr{O}_D\to \Omega_D(0)$? More generally, suppose $\nabla$ is a holomorphic connection on $\mathscr{O}_U^{\oplus r}$, can we extend it to a logarithmic connection on $\mathscr{O}_D^{\oplus r}$. |
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Connections on the punctured disk extend (up to isomorphism) to connections with log poles, but not in general to holomorphic connections. A standard reference is: Pierre Deligne. "Equations differentielles a points singuliers reguliers. Springer-Verlag, Berlin, 1970. Lecture Notes in Mathematics, Vol. 163. I'd say the key point is that in one dimension the connection is integrable corresponds to a local system defined by its monodromy - an $r\times r$ matrix A, which will have a logarithm. Perhaps someone else can suggest a more recent exposition. |
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