[*Edit (June 20, 2010):* I posted [an answer](https://mathoverflow.net/a/28891/14094) to this which summarizes one that I received verbally a few weeks after posting this question. I hope it is useful to someone.]


I am presently seeking references which introduce "formal geometry". So far as I can tell, this idea was presented by I.M. Gel'fand at the ICM in Nice in 1970. There is his lecture, a paper by him and Fuks, and also a paper by Bernshtein and Rozenfeld with some applications that I don't understand too well. What I am unable to find is a thorough exposition of the foundations. It seems like a canonical enough construction that it should have been included in some later textbook, (though apparently not called "formal geometry" since that is not turning up anything useful).


Below is what I understand, which is several main ideas, but missing many details; this will most certainly be riddled with errors, because I am only able to give what I have roughly figured out from reading incomplete (though well-written and interesting!) sources, and asking questions. I am including it in the hopes it will be familiar to some kind reader.



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> I'm sorry to not ask a specific question. Hopefully some answers will help me edit the below description to remove inaccuracies, and some others will suggest references. Both would be very helpful.
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Let $X$ be a smooth complex algebraic variety of dimension $n$ (could just as well be a complex analytic or smooth real manifold so far as I understand; probably can be algebraic over any field, at least for awhile). There is a completely general torsor over $X$: its fiber over a point $x$ is the set of all coordinate charts on the formal neighborhood of $x$ in $X$. This is a torsor over the infinite dimensional group G of algebra endomorphisms of $\mathbb{C}[\![x\_1,...,x\_n]\!]$ which preserve the augmentation ideal and are invertible modulo quadratic terms (and hence invertible over power series of endomorphisms). It's a torsor because any two coordinate systems are related by such an endomorphism, but there isn't a canonical choice of coordinate system along the variety.


I think one can rephrase the conditions of the previous paragraph more precisely by first noting that an endomorphism of $\mathbb{C}[\![V]\!]$ preserving augmentation ideal (where we use notation $V=\operatorname{span}\_\mathbb{C}(\{x\_1,...,x\_n\})$ is given by a linear map $V\to V\ast \mathbb{C}[\![V]\!]$, which then uniquely extends to an algebra map. Then the condition of the last paragraph is that
$$V \to V\ast \mathbb{C}[\![V]\!]\to V\ast \mathbb{C}[\![V]\!] / V\ast V\ast \mathbb{C}[\![V]\!] = V$$ is invertible.


It's not hard to see that these in fact form a group, and that this group acts simply transitively on the set of coordinate systems.


The Lie algebra $\mathfrak{g}$ of $G$ (once one makes sense of this) is a subalgebra $W^0$ (described below) of the Lie algebra $W\_n$ of derivations of $\mathbb{C}[\![x\_1,...,x\_n]\!]$. $W\_n$ is the free $\mathbb{C}[\![x\_1,...,x\_n]\!]$-module generated by $\partial\_1,\dots,\partial\_n$, with the usual bracket.


$W=W\_n$ has a subalgebra $W^0$ of vector fields which vanish at the origin (i.e. constant term in coefficients of ∂i are all zero), and another $W^{00}$ of vector fields which vanish to second order (so constant and linear terms vanish). It's fairly clear that $W^0/W^{00}$ is isomorphic to $\mathfrak{gl}\_n$. One now considers W\_n modules which are locally finite for the induced $\mathfrak{gl}\_n$-action. It turns out that these can be "integrated" to the group $G$, because $G$ is built out of $\operatorname{GL}\_n$ and a unipotent part consisting of those endomorphisms which are the identity modulo $V\ast V$. So the integrability of the $\mathfrak{gl}\_n$-action is all one needs to integrate to all of $G$.


Now one performs the "associated bundle construction" in this context, to produce a sheaf of vector spaces out of a W\_n module of the sort above. One could instead start with a f.d. module $V$ over $\mathfrak{gl}\_n$, and there's a canonical way to turn it into a $W\_n$-module (in coordinates you tensor it with $\mathbb{C}[\![x\_1,...,x\_n]\!]$ and take a diagonal action: W\_n acts through $\mathfrak{gl}\_n$ on the module $V$ and by derivations on $\mathbb{C}[\![x\_1,...,x\_n]\!]$). The sheaves you get aren't a priori quasi-coherent; some can be given a quasi-coherent structure (i.e. an action of the structure sheaf on X) and some can't. However, the sheaves you get are very interesting. By taking the trivial $\mathfrak{gl}\_n$-bundle you get the sheaf of smooth functions on the manifold (this was heuristically explained to me as saying that to give a smooth function on a manifold is to give its Taylor series at every point, together with some compatibilities under change of coordinates, which are given by the $W\_n$-action). By taking exterior powers of $\mathbb{C}^n$ you recover the sheaves of differential forms of each degree (these examples can be made into quasi-coherent sheaves in a natural way). The $W\_n$-modules associated to the exterior powers are not irreducible; they have submodules, which yield the subsheaves of closed forms (these give an example of a sheaf built this way which isn't quasi-coherent: function times closed form isn't necessarily closed).


Finally, one is supposed to see that the existence of the extra operators $\partial\_i$ of W which aren't in $W^0$ further induce a flat connection on your associated bundle. I don't yet understand the underpinnings of that, but it's very important for what I am trying to do.



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> Is this familiar to any readers? Is there a good exposition, or a textbook which discusses the foundations? Can anyone explain the last paragraph to me?
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Formal geometry
This was briefly discussed [previously on this site](https://mathoverflow.net/questions/518/what-is-the-connection-between-d-modules-and-coordinate-bundles/523). You can find discussions of it in:


1. Chapters 6 and 17 of Frenkel and Ben-Zvi: *Vertex algebras and algebraic curves.*
2. Section 2.6.5 of Beilinson and Drinfeld's *Quantization of Hitchin's Integrable system* (available online)
3. Section 2.9.9 of Beilinson and Drinfeld's *Chiral Algebras*.


