Extending holomorphic connections - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T13:47:08Z http://mathoverflow.net/feeds/question/54832 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/54832/extending-holomorphic-connections Extending holomorphic connections Rex 2011-02-08T22:55:22Z 2011-02-08T23:13:08Z <p>Let $D$ denote the disk $|z|&lt;1$ 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 $\nabla_D:\mathscr{O}_D\to \Omega_D(0)$?</p> <p>More generally, suppose $\nabla$ is a holomorphic connection on $\mathscr{O}_U^{\oplus r}$, can we extend it to a logarithmic connection on $\mathscr{O}_D^{\oplus r}$.</p> http://mathoverflow.net/questions/54832/extending-holomorphic-connections/54833#54833 Answer by profilesdroxford54 for Extending holomorphic connections profilesdroxford54 2011-02-08T23:13:08Z 2011-02-08T23:13:08Z <p>Connections on the punctured disk extend (up to isomorphism) to connections with log poles, but not in general to holomorphic connections. </p> <p>A standard reference is: Pierre Deligne. "Equations differentielles a points singuliers reguliers. Springer-Verlag, Berlin, 1970. Lecture Notes in Mathematics, Vol. 163.</p> <p>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.</p> <p>Perhaps someone else can suggest a more recent exposition.</p>