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Initially I wanted to call this question "Categorification of meromorphic functions?" but discovered so many questions about categorification that I became scared and decided to replace it with a more obscure title.

The question resulted from contemplating my own answer to Modern treatment of Dirac monopoles and related topics.

Let us work with some version of the notion of variety, say, complex varieties. Let $V$ be one such.

Given, say, a regular function on $V\setminus\{\text{a point}\}$, this function might have a pole of some order at that point, or an essential singularity. This is absolutely classical, one way to deal with it is to understand regular functions as morphisms to $\mathbb A^1$ and meromorphic functions as morphisms to $\mathbb P^1$.

Slightly later: as pointed out by Alexandre Eremenko in a comment, what I said above needs correction. Inverse image of infinity under a morphism to $\mathbb P^1$ only can reduce to a finite set of points when $V$ is 1-dimensional; for general $V$ it will have codimension 1. Whereas for morphisms to $\mathbb P^N$ with $N$ large considered later I am not sure what happens. Still let me leave the rest as is for the time being.

Suppose given a line bundle on $V\setminus\{\text{a point}\}$. What can happen at that point?

The analogy with functions might go like this. View "regular" line bundles as morphisms to $\mathbb P^N$ for large enough $N$, or maybe to $\mathbb P^\infty$ which is defined as an ind-scheme obtained from the union of all the $\mathbb P^N$ in a certain way.

Is there some $\mathbb X$ that is to $\mathbb P^\infty$ as $\mathbb P^1$ is to $\mathbb A^1$? That is, such that $\mathbb P^\infty$ maps to $\mathbb X$, representing a hypothetical map (functor?) from "regular" line bundles to "meromorphic" line bundles?

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    $\begingroup$ You can certainly define a meromorphic line bundle as one where the transition functions can have zeros or poles along the divisor. Not sure what the classifying space for these would be in the sense you ask for $\endgroup$
    – Kevin Casto
    Commented Mar 16, 2023 at 10:34
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    $\begingroup$ @KevinCasto Thank you, this sounds promising. Can the ones you mention be described by some kind of cohomology group? Do they have algebraic duals, analogous to fractional ideals or rank 1 projective modules? $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Mar 16, 2023 at 11:07
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    $\begingroup$ Sure, they'll just be $H^1$ of $V$ with coefficients in the sheaf of transition functions I described, but that's kind of tautological. $\endgroup$
    – Kevin Casto
    Commented Mar 16, 2023 at 11:14
  • $\begingroup$ You mean sheaf of functions to $\mathbb P^1$? But it does not have any group structure, does it? $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Mar 16, 2023 at 11:33
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    $\begingroup$ "Given, say, a regular function on V∖{a point} this function might have a pole of some order at that point, or an essential singularity" - are you talking about 1-dimensional $V$? In higher dimension a function regular on $V$ minus {point} cannot have a pole. $\endgroup$
    – Alexandre Eremenko
    Commented Mar 16, 2023 at 14:01

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You can define "meromorphic vector bundle" as locally free sheaf of modules over a sheaf of meromorphic functions. This is a highly non-trivial object, because (in contrast with rational functions) meromorphic functions might have non-trivial first cohomology. When your manifold is very non-algebraic (such as a general K3 surface), the category of reflexive sheaves is embedded to the category of meromorphic vector bundles as an abelian tensor subcategory, but it's not clear if all meromorphic vector bundles are obtained this way (I guess not).

In dimension 1 everything can be computed explicitly, and nothing is different from rational functions, but in dimension 2 differences start. It seems that the category of meromorphic vector bundles is non-trivial even on ${\Bbb C}^2$.

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  • $\begingroup$ Thank you! May I ask this: my main motivating example (which I probably should include in the question) was a passage from "Loop spaces, characteristic classes and geometric quantization" by Brylinski, where he proposes to interpret Dirac's magnetic fields with a singularity at the origin by line bundles on $\mathbb R^3\setminus\{0\}$. I was wondering what happens if one tries to extend such bundle to the whole $\mathbb R^3$ - seemingly certain cohomology class serves as an obstruction for that. So, what is this? Some sheaf on $\mathbb R^3$ or..? $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Apr 3, 2023 at 5:36

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