Formal geometry

Question

[Edit (June 20, 2010): 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 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.

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?

Answer 1

This was briefly discussed previously on this site. 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.

Answer 2

I am writing to post an answer to my own question. The answer below consists of what I was able to jot down from a seminar talk by Jacob Lurie, and a patient followup explanation by Roman Travkin, followed by a correction by David Ben-Zvi. Of course, mistakes and naivete in the translation are solely attributed to me. Since the answer was given to me in response to my asking on MO, it seems that karma dictates that I record it here.

The deRham stack, $X_{dR}$, of a scheme $X$ is defined as a functor on test schemes S by

$$X_{dR}(S) := \{\textrm{Maps}: S_{red} \to X \}.$$

where $S_{red}$ is the reduced scheme associated to a scheme $S$. It is representable as a stack - I think always, but at least when $X$ is smooth - which we should assume for later purposes anyways (maybe everything should be based over $\mathbb{C}$ also? I welcome corrections, which I'll incorporate).

Okay so $X$, viewed as the functor $X(S)=\{\textrm{Maps}:S\to X\}$ has a natural map $\pi:X\to X_{dR}$, by pre-composing with $S_{red} \to S$.

There is a scheme $Aut_X$ of infinite type over a smooth $X$ whose set of points consists of pairs $(x\in X, t_x: X_{(x)}\cong \hat{D_n})$, where $\hat{D_n}$ is the formal disc, $\mathcal{O}(\hat{D_n}):=\mathbb{C}[[x_1,\ldots,x_n]]$. The fiber over each point $x\in X$ is the group $Aut^0(X_{(x)})$ of automorphisms of the formal neighborhood of $x$ preserving the maximal ideal, which is in turn (non-canonically) isomorphic to $Aut^0(\hat{D_n})$, the group of automorphisms of $\hat{D_n}$ preserving the origin.

Now, the scheme $Aut_X$ is actually a $\hat G$-torsor over $X_{dR}$, where (I gather from David Ben-Zvi's comments) $\hat G$ is something more like an inductive limit of the groups of automorphisms on $n$th infinitessimal neighboroods of the origin in $D_n$.

Now, suppose that $M$ is a $W_n$ module, which has the property that the operators $x_i\partial_i$ for all $i$ are diagonalizable with finite dimensional eigenspaces. (These we should think of as being the diagonals $h_i$ for a copy of $\mathfrak{gl}_n$ sitting inside $W_n$). In this case, one can prove some nice things.

First, in representation theory: $M$ lies in the category generated by modules $\mathcal{F}_\lambda$ which are coinduced from $V_\lambda$, the irreducible of $\mathfrak{gl}_n$.` That is, as vector spaces the $\mathcal{F}_\lambda$s are just $V_\lambda \otimes C[x_1,\ldots, x_n]$, and the action is given by natural formulas. So $M$ is a (possibly infinite) extension of such things. But there is a theory of weights which control the representation theory somewhat. See A. N. Rudakov. Irreducible representations of infinite-dimensional Lie algebras of Cartan type. Math. USSR Izv. Vol. 8, pgs. 836-866, which has been translated to English.

Second, in geometry: The module $M$ is a Harish-Chandra module for the pair $(W_n,Aut_0(C[[x_1,...,x_n]]))$. This means: $W_n$ has a Lie sub algebra $W^0$ of vector fields which vanish at the origin (so they have no constant vector field terms like $\partial_i$). $W^0$ (or perhaps its completion w.r.t order of vanishing at origin) is the Lie algebra of $Aut_0(C[[x_1,...,x_n]])$, as it consists of derivations which preserve the maximal ideal (not sure exactly how you make precise that it is "the Lie algebra", but anyways, there is an exponential map turning an integrable $W^0$ module into an $Aut_0(C[[x_1,...,x_n]]$)-module ). The assumptions on $M$ were precisely those that make $M$ an integrable $W^0$ module, and so we can regard $M$ as a $G$-module. Well now we have an associated bundle construction for the $G$-torsor $Aut_X\to X$, and we can use this to produce a sheaf of vector spaces (not quasi-coherent!) over $X$ with fiber $M$.

Better $M$ has an action of the operators $\partial_i$, and we can exponentiate the action of all of $W_n$ to the group $\hat G$. That means that we can instead construct the associated bundle for the $\hat G$ torsor $Aut_X\to X_{dR}$, which will give us a bundle over $X_{dR}$ with fiber $M$ again. Now, we can pullback this bundle via $\pi$ to get a bundle on $X$ with fiber $M$. This time, however, it's a pullback of a sheaf of vector spaces on the deRham stack, which by (some people's) definition is a crystal of vector spaces on $X$.

Answer 3

Some ideas of references that come to my mind right away:

• Fuks, Cohomology of infinite-dimensional Lie algebras. - contains lots of things that are relevant to your question
• Gelfand and Kazhdan, Certain questions of differential geometry and the computation of the cohomologies of the Lie algebras of vector fields.
• Gelfand, Kazhdan and Fuks, Actions of infinite-dimensional Lie algebras.
• a mysterious preprint of Guillemin, "Notes on Gelfand-Fuks cohomology" (I have never seen it but it is supposed to be available through the MIT library - and is supposed to be very good :) )

Also, you might want to look at Kontsevich's paper "Formal (non)commutative symplectic geometry."

Sorry if you know of all these already.

Answer 4

Have you looked at Bezrukavnikov and Kaledin's Fedosov quantization in algebraic context? They say specifically that they couldn't find any good references and thus had to write up several things themselves.