MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 $\mathscr{O}_D^{\oplus r}$.

share|cite|improve this question

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.

share|cite|improve this answer
The book Isomonodromic deformations and Frobenius manifolds by Sabbah deals with that. – Sebastian Feb 9 '11 at 7:33

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.