Timeline for A de Rham space for meromorphic connections?
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 26, 2021 at 12:15 | vote | accept | Pulcinella | ||
| Jan 26, 2021 at 12:14 | comment | added | Pulcinella | I had made a typo, but in correcting it I realised that you are right, algebraic meromorphic connections are the same thing as connections on an open set. I think I had assumed that it was possible to define de Rham space in the analytic setting somehow, but I also have no idea how to do this. | |
| Jan 26, 2021 at 0:24 | comment | added | dhy | @Meow I am now very confused. Are you working in the algebraic or the analytic world? If you are working in the analytic world then I agree with what you are saying, but then I don't know how to make sense of even the normal de Rham stack $X_{dR}$. If you are working in the algebraic world then I see no difference between meromorphic connections and $D$-modules on $U$. (In particular, $\mathcal{O}_X(D)$ and $\mathcal{O}(U)$ coincide in the algebraic world no?) | |
| Jan 25, 2021 at 17:57 | comment | added | Pulcinella | A meromorphic (flat) connection on $X$ is a $\mathcal{O}_X(D)$ module $M$ (with $D$ a divisor) along with a map $\nabla : M\to M\otimes_{\mathcal{O}_X}\Omega^1_X$ with Liebnitz (and flatness). e.g. Hotta et al's book on D modules, 5.2.1. That is, you genuinely care about the extra structure at the singularity. A connection on $U\subseteq X$ might not extend/extend uniquely to a meromorphic connection on $X$. This is true even in the regular singular case, where the extension is not unique. | |
| Jan 24, 2021 at 23:55 | comment | added | dhy | @Meow Why is it not? | |
| Jan 24, 2021 at 22:59 | comment | added | Pulcinella | The problem with that is that vector bundles on a pushout of $Z\to Y_i$ are vector bundles on $Y_i$ restricting to the same bundle on $Z$. So in your case vector bundles on the pushout will be a vector bundle on $X$ along with a holomorphic connection on some open subset of $X$. This is closer to the correct answer, but not quite there. | |
| Jan 23, 2021 at 22:01 | history | bounty ended | CommunityBot | ||
| Jan 20, 2021 at 0:49 | comment | added | dhy | @Meow I wanted to take the pushout along the maps $\eta\rightarrow X,X_{mdR}$. Sorry, I got distracted; I'll try to make some time to think about this tomorrow. | |
| Jan 17, 2021 at 13:36 | comment | added | Pulcinella | What would the maps $X_{mdR}\to X,\eta$ be? I agree it feels like something like this should work. | |
| Jan 16, 2021 at 13:37 | comment | added | dhy | Ah OK, I misunderstood. I assumed by meromorphic connection you simply meant a $\mathcal{D}$-module which is intermediate extended, but you actually want that plus the data of a lattice. Let me think and see if I can say anything about that case. I am tempted to just take the coproduct of $X_{mdR}$ and $X$ over $\eta$ - let me think if that works... | |
| Jan 16, 2021 at 13:31 | comment | added | Pulcinella | Are you claiming that vector bundles on your prestack are the same thing as vector bundles with connection on an open subset of $X$? The concern I would then have is that these do not have a unique extension to a meromorphic connection on all of $X$ (e.g. by Deligne's theorem). | |
| Jan 16, 2021 at 7:07 | history | answered | dhy | CC BY-SA 4.0 |