[edit: I posted an answer 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.

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.

Let X be a smooth complex algebraic variety of dimension n (could just as well be an 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 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 C[[V]] preserving augmentation ideal (where we use notation V=span_C{x_1,...,x_n}) is given by a linear map V-->V*C[[V]], which then uniquely extends to an algebra map. Then the condition of the last paragraph is that V -->V*C[[V]]-->V*C[[V]] / V*V*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 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 C[[x_1,...,x_n]]. W_{n} is the free C[[x_1,...,x_n]] module generated by ∂_{1},..∂_{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 gl_n. One now considers W_n modules which are locally finite for the induced gl_n action. It turns out that these can be "integrated" to the group G, because G is built out of GL_n and a unipotent part consisting of those endomorphisms which are the identity modulo V*V. So the integrability of the 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 gl_n, and there's a canonical way to turn it into a W_n module (in coordinates you tensor it with C[[x_1,...,x_n]] and take a diagonal action: W_n acts through gl_n on the module V and by derivations on 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 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 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 ∂_{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.

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?

arenow available online: mathunion.org/icm/proceedings . $\endgroup$ – LSpice Jul 5 at 6:24