# connections with regular singularities

Let $k$ be a field of characteristic zero, $X=\mathbb{G}_{m, k}=\mathrm{Spec}\ k[t, t^{-1}]$ the multiplicative group over k and $E=\mathcal{O}_X$ the trivial line bundle.

Consider the connection $\nabla: E \to E \otimes_{\mathcal{O}_X} \Omega^1_X$ defined by $\nabla(1)=\alpha \frac{dt}{t}$ for some $\alpha \in \mathbb{C}$, that is:

$\nabla=d+\alpha \frac{dt}{t}\wedge$

My question is: has $(E, \nabla)$ regular singularities?

Thanks for you help

• p^4: I didn't downvote you and yes it does. Feb 5, 2013 at 16:19
• Dear Donu, could you explain why ? It seemed pretty obvious to me but then I found the paper mi.fu-berlin.de/users/esnault/preprints/helene/… page 2 and I was confused about the statement on sections
– pppp
Feb 5, 2013 at 16:39
• It's obvious if you can use the standard Fuchs criterion that the singularities of the connection are logarithmic (cf Deligne's book). I get $t^s$ as a basis of horizontal sections in the example in the linked preprint. I'm sure they mean something else, but I don't have the time to read it carefully right now. Feb 5, 2013 at 16:57
• That's the point. I don't see what exp(-t) has to do with horizontal sections
– pppp
Feb 5, 2013 at 17:43

Dear $p^4$,
I had a chance to glance further at the 4 author paper linked in your comment. Their connection form is $-dt + sdt/t$. This has double pole at $\infty$, so it isn't regular*. However, the connection in your question has logarithmic singularities at $0$ and $\infty$, so it is regular, as I said earlier. There is no contradiction.
*(Afterthought) This is a bit sloppy, since the Fuchs criterion is only a sufficient condition for regularity. But to see the non regularity, observe that the solution they write down $exp(t)t^{-s}$, which is correct (!), has bad singularities at $\infty$.