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JMRA Video

6th of September, 2020

I recently submitted a video to the Junior Mathematician Research Archive that gives a brief overview of the work in my PhD thesis. You can watch the video here but, be warned, it’s really probably not the most coherent (ha ha) narrative.

For the sake of it, or in case you’re the sort of person who prefers to read instead of watch, I’ve included the transcript of the whole video below.

Hi, my name is Tim Hosgood, and I’m a PhD student at the University of Aix-Marseille, currently finishing a year-long position as part of the DerSympApp team at the University of Montpellier.

This video, made for the Junior Mathematician Research Archive, aims to explain the work of my thesis, which was split into two papers, preprints of which can be found on the arXiv under the titles “Simplicial Chern-Weil theory for coherent analytic sheaves”, Parts I and II. For this video though, I’m going to be working from my thesis, just because then everything is in one place instead of two.

I’m going to assume that you have at least a passing familiarity with most of the following ideas: manifolds, vector bundles, connections and their curvatures, sheaves, some homological algebra, and an example or two of simplicial objects (that is, contravariant functors from the abstract simplex category into some target category). The intended audience is really anybody who cares about either Chern-Weil theory or coherent analytic sheaves.

Let’s start with some historical context.

In 1980, H.I. Green wrote his thesis on the subject of Chern classes of coherent sheaves on complex-analytic manifolds. It was never published, but an exposition was written by Toledo and Tong, whose work on twisting cochains with O’Brian (one of Green’s supervisors) formed the backbone of Green’s thesis, alongside Dupont’s fibre integration. The construction, which gives classes in \mathrm{H}^{2k}(X,\Omega_X^{\bullet\geqslant k}), agrees with the classical construction by Atiyah-Hirzebruch in \mathrm{H}^{2k}(X,\mathbb{Z}).

Then, nearly 30 years later, in Grivaux’s thesis, this construction was mentioned as one that should fit into an axiomatisation of Chern classes on (compact) complex-analytic manifolds that ensures uniqueness, but the details still hadn’t quite been worked out.

Finally (for now, at least, but hopefully not forever), this year we figured out how to fit the pieces together, by showing that Green’s construction is a specific (and prototypical) example of a more abstract construction, and, furthermore, satisfies the axioms of Grivaux in the compact case.

For those who work with Chern classes in algebraic geometry, this might all seem rather odd. Indeed, in the complex algebraic world, the construction of Chern classes has been very well understood since work by Grothendieck (amongst others). But there are two problems with using Chern-Weil theory in the complex analytic world:

  1. coherent sheaves often do not admit global resolutions by locally-free sheaves;
  2. global holomorphic connections on holomorphic vector bundles very rarely exist.

This means that the classical method (take a locally-free resolution, take connections on each vector bundle, use Chern-Weil on the curvature of each connection, and then take an alternating product) breaks down in two ways. Luckily though, both of these problems are sort of the same: they result from the non-existence of a global object, even though local objects always exist.

That is, we always have local resolutions of coherent analytic sheaves by locally-free sheaves, and local holomorphic connections always exist on holomorphic vector bundles. This is where simplicial methods become incredibly powerful: we actually can obtain some global object if we slightly weaken what we mean by “global object”. But then, using Dupont’s fibre integration, we can actually turn these “quasi-global” (i.e. simplicial) objects back into strictly global ones.

So here’s the plan:

  1. figure out how to do Chern-Weil theory for vector bundles, i.e. look at simplicial connections;
  2. try to understand how coherent analytic sheaves are resolved by so-called “vector bundles on the nerve” (which are somehow the “good” notion of “simplicial vector bundles”, even though they are slightly different from what this name may suggest);
  3. prove that everything fits together well homotopically, i.e. on the level of (\infty,1)-categories, and that we recover all the classical calculations.

So let’s start with some maths. One fundamental construction is that of the Čech nerve of a cover, which is a simplicial space, i.e. a collection of spaces, one for each p\in\mathbb{N}, along with face and degeneracy maps between them. This is something that we will be using a lot in what follows, and we can think of it as a sort of “exploded version” our a space.

For what follows, let’s fix some paracompact complex-analytic manifold X with a finite Stein cover \mathcal{U}=\{U_\alpha\}, and take some locally free sheaf E (which we can also think of as a holomorphic vector bundle) of rank \mathfrak{r}. For simplicity, we assume that \mathcal{U} trivialises E. So we have transition maps M_{\alpha\beta}, which can be realised, if we take some local connections s_\alpha, as an (\mathfrak{r}\times\mathfrak{r})-matrix.

We then consider the Atiyah exact sequence of \mathcal{O}_X-modules (containing the first jet bundle) and the corresponding \mathrm{Ext}-class, which is called the Atiyah class. It is a nice classical fact that the trace of this class somehow “is the same as” the first Chern class, but this class also describes the obstruction towards E admitting a global holomorphic connection. Indeed, a connection is exactly a the data of a splitting of this short exact sequence.

So already we see how Chern-Weil theory “breaks down”: the moment our vector bundle has a non-trivial first Chern class, it cannot admit a global holomorphic connection, so we can’t take the curvature and evaluate some invariant polynomial on it. That is, the moment there is a first Chern class worth calculating, we can’t calculate it with Chern-Weil theory!

But let’s be a bit more explicit about this Atiyah class: assuming that we have local holomorphic connections (which is always the case if, e.g. we have a basis of local sections), we can actually write down the Čech cocycle in terms of the transition functions of E, and the trace of this is de Rham closed (and, by definition as a Čech cocycle, also Čech closed), which means that it gives us a closed class in the total complex of the Čech-de Rham bicomplex. Since we are working in a nice enough setting, this means that we get a class in de Rham cohomology (or even truncated de Rham cohomology).

Before introducing any of the formalism of simplicial constructions (i.e. these “sheaves on the nerve”), we can think about the more computational problem: can we lift the kth Atiyah class (which is just the wedge product of k copies of the Atiyah class) to give a closed element in the Čech-de Rham total complex when k>1?

The answer is yes! For the second Atiyah class, by calculating the image under the de Rham differential, and considering all possible polynomials in one formal variable and their images under the Čech differential, and working with polynomials in two formal variables up to cyclic permutation (under which the trace is invariant), we can brute-force equate coefficients, and find a Čech 1-cocycle of 3-forms such that we get a closed element in the total complex.

We can do the exact same process, but twice, to compute the analogous “correction terms” for the third Atiyah class, but already we see that things start to get a bit messy. By the time we reach the fourth Atiyah class, the polynomials are already too large to be able to spot a pattern by just looking at them, and so we see that we need a slightly fancier approach if we want any hope of getting classes in de Rham cohomology for further Atiyah classes.

This is where we now turn to the simplicial construction. First we need to know what the “good” notion of a simplicial vector bundle (or, more generally, a simplicial sheaf) is. This is something that was described by Green, but hasn’t seen much use since. In fact, these objects are interesting enough in their own right that they deserve a more thorough treatment, since they can actually be realised by an abstract (but concrete) construction of Bergner, as a lax homotopy limit of model categories. But that’s a story for another time; here we’ll just give the definition, as well as a vague motivation. A sheaf on the nerve is a collection of sheaves, indexed by p\in\mathbb{N}, with each \mathcal{E}_p being a sheaf on the pth level of the Čech nerve, such that, for any morphism in the abstract simplex category, we get (covariantly) a functorial morphism of sheaves. Here, “functorial” means that it respects composition, and when we say “sheaves” we can mean just topological sheaves, or (what we’re more interested in) sheaves of \mathcal{O}_X-modules, but really each \mathcal{E}_p will be a sheaf of \mathcal{O}_{X_p}-modules. These sheaves on the nerve have sometimes been called “simplicial sheaves”, but this is a misleading name, since they’re really not simplicial objects in some category of sheaves (in fact, they’re not even simplicial, but instead cosimplicial, and even then they’re not just cosimplicial objects in some category!), so we stick to saying “sheaves on the nerve”, even if it is a bit longwinded.

An important, and even prototypical, example of a sheaf on the nerve is given by pulling back a sheaf on X along the map from the Čech nerve to X. So now we can state our current goal a bit more explicitly: if we take a holomorphic vector bundle with non-trivial Chern class (and thus does not admit a global holomorphic connection), pull it back to the nerve to get a vector bundle on the nerve (i.e. a locally free sheaf on the nerve), is there some good notion of “simplicial connection” such that (a) these things always exist, and (b) we can take their curvatures, evaluate them under invariant polynomials, and get some class in de Rham cohomology?

The answer to this question is, luckily, yes, and this was exactly one of the two main things that Green showed in his thesis (the other being the existence of very nice simplicial resolutions of coherent sheaves, which we will come to later). But Green’s construction, while perfectly functional, does not really give any indication as to why this method works, nor how we could generalise it to give a general definition of a simplicial connection. That is the aim of this chapter of my thesis.

Although there are a lot of very technical (and seemingly uninviting) proofs, the main motivation, which can be found in Green’s thesis, is really quite simple and beautiful. It goes as follows. Locally, we always have homolorphic connections, say \nabla_\alpha on each U_\alpha. We know that we cannot glue them together if we have a non-trivial first Chern class, i.e. that, on intersections U_{\alpha\beta}, it will not be the case that \nabla_\alpha and \nabla_\beta agree. But what we can do then is just take both of them, in a barycentric average, if you will. That is, we consider (\lambda-1)\nabla_\alpha+\lambda\nabla_\beta for \lambda\in[0,1]. When \lambda=0 we recover \nabla_\alpha and when \lambda=1 we recover \nabla_\beta, and for any other value of \lambda we get some weighted average of the two. So what we’ve constructed on double intersections is a connection parametrised by [0,1], i.e. by the 1-simplex! We can do the same on triple intersections: take some linear combination of \nabla_\alpha, \nabla_\beta, and \nabla_\gamma, where we parametrise by the 2-simplex. We can think of this as a triangle, and then any point on the boundary corresponds to forgetting one of the local connections, and any vertex corresponds to forgetting two of them, and just remembering one. Proceeding like this, we can build, on each level of the Čech nerve (i.e. on each p-intersection), a connection that is parametrised by the p-simplex.

Using this construction, we can explicitly calculate the curvature, which gives us what we call the first simplicial Atiyah class, since we can show that, working over U_{\alpha_0} on any p-intersection U_{\alpha_0\ldots\alpha_p}, it contains the Čech cocycle representing the Atiyah class that we’ve already calculated. If we then take the trace of this, we get what is called a simplicial differential form, i.e. a bunch of differential forms on the simplex times the Čech nerve that satisfy some gluing condition resembling that for the fat geometric realisation. But this is where Dupont’s fibre integration comes into play: there is a quasi-isomorphism between the simplicial de Rham complex and the usual de Rham complex. So we can go through and write down what we get when we follow this whole procedure, and we see that the fibre integral of the trace of the first simplicial Atiyah class is exactly the trace of the first Atiyah class! In fact, the same is true for the second Atiyah class too, and for the third (although now the equality only holds in cohomology, and not directly on the level of cocycles). More generally, as one might hope, we can show that the result is true for all the Atiyah classes.

The above is really just a nice way of explaining in more detail what Green did in his thesis, but we said that we were going to do something a bit more general, and give abstract definitions for simplicial connections, so let’s do that now.

We already have a vague idea of what a simplicial connection should be: it should be a bunch of connections, each one on a pth level of the Čech nerve times the p-simplex. But we will probably need to be a little bit more strict with what we call a simplicial connection, since this object seems pretty general.

Indeed, let’s think about what we need the connection to satisfy.

First, we want to get a simplicial differential form when we take the trace of the curvature, so that we can apply Dupont’s fibre integration. This means that the simplicial connection should satisfy some sort of compatibility between different simplicial levels, akin to the equation satisfied by simplicial differential forms. This can be guaranteed by asking for a certain condition on what we call the comparison maps. That is, we say that such a collection of connections is a simplicial connection if the comparison maps are true morphisms, which means that they commute with the connections. Again, this seemingly arbitrary condition is actually exactly what we need to ensure that the resulting differential forms will satisfy the simplicial condition needed to be able to apply Dupont’s fibre integration quasi-isomorphism.

Next, we need to worry about the fact that we have a bunch of connections, and so, when we take their curvatures and evaluate them under some invariant polynomials (such as the trace, after taking a wedge product), there is no reason a priori that we will get the same result at each simplicial level, i.e. we might get a bunch of different characteristic classes. We can solve this problem by asking for the comparison maps to be admissible, which is a purely formal property motivated by the following simple (but entirely arbitrary-seeming, at this point) definition: given pairs (V,\varphi) of finite-dimensional vector spaces and endomorphisms on those spaces, we say that a morphism f\colon(V,\varphi)\to(W,\psi) (assumed a priori to commute with the endomorphisms) is admissible if there exists subbundles V_1\hookrightarrow V and W_1\hookrightarrow W such that f restricts to a morphism V_1\to W_1 and further descends to an isomorphism V/V_1\xrightarrow{\sim}W/W_1. In some sense (which we can make much more precise), a morphism is admissible if it is an isomorphism away from some “flat” subbundles.

If all this sounds a bit hand-wavy, then it’s probably best to go and read this in some more detail, because I don’t want to get lost in all the technical details here, and there really isn’t anything that fancy going on. But I promise that this simple idea of “admissibility of a simplicial connection is exactly the condition needed to ensure that the characteristic classes are well defined” is all that’s going on behind the scenes.

Finally, we might worry about our initial data, i.e. if we started off with some different choices of local sections and local connections, then would we still end up with the same characteristic classes? The answer is, a priori, no, so we need to ask for another condition. We say that a set of (admissible) simplicial connections is compatible if the difference between any two of them is itself an admissible endormorphism-valued simplicial 1-form. We haven’t defined here what exactly such an object is, but it’s written down in full detail, and, again, the idea is exactly the same as above: it’s something that is somehow isomorphic between different simplicial levels, if you quotient out by some sub-part that is, in some sense, “flat”.

Note also that we haven’t really said what an invariant polynomial is, since we actually need to modify the classical definition slightly, because we need a bunch of invariant polynomials: one for each simplicial level. Again, there are some small technical conditions that we impose on them, but they aren’t very restrictive, and all the things that you would hope to be invariant polynomials (such as the trace precomposed with the wedge product (or composition)) are indeed “generalised” invariant polynomials.

The last, very reassuring result, of this section is the following: Green’s “barycentric” connection, whose construction we described earlier, is indeed an admissible simplicial connection. In fact, an even stronger result holds: given a sufficiently nice complex of vector bundles on the nerve (where here “sufficiently nice” will be explained when we talk about Green’s resolution), we can endow each vector bundle on the nerve with an admissible simplicial connection, and the family of all such connections arising from different initial data is a compatible family. That is, if we have a nice enough complex of vector bundles on the nerve, then we can do Chern-Weil theory to calculate Chern classes of each of them. To prove this, we introduce the idea of a simplicial connection being “generated in degree zero”, which, very loosely, but very accurately, means “looking exactly like Green’s barycentric connection”.

The second piece of the construction that we’ve been skirting around is how to bring this construction up to the level of coherent sheaves, i.e. how to get around the problem of coherent analytic sheaves not having global resolutions by locally free sheaves. This is something else that Green solved in his thesis, by using the twisting cochains of Toledo and Tong (very linked to, but not quite the same as, the twisting complexes of Bondal and Kapranov) to prove the following result: any coherent analytic sheaf, when pulled back to the nerve, has a resolution by vector bundles on the nerve, and these vector bundles satisfy some nice properties. What are the nice properties satisfied by these vector bundles? Well, one way of saying it is the following: they are exactly the nice properties that we said we would need to prove our previous theorem, about being able to endow a complex of vector bundles on the nerve with admissible simplicial connections in a “compatible family” way. Another way of describing them is the following: on any intersection U_{\alpha\beta}, these vector bundles on the nerve look like the part over U_\alpha direct summed with something that is “homotopically trivial” (formally speaking, something that is elementary, i.e. a direct sum of (possibly shifted) identity morphisms). This means that the coface maps are injections, and that the codegenacy maps are surjections. Recalling what we said quite a few minutes ago, although sheaves on the nerve are not cosimplicial objects in some category of sheaves, when they satisfy this property, they look a lot closer to such things, since they have injective coface maps, which is something that cosimplicial objects always have!

Twisting cochains, and, indeed, Green’s resolution, come with a lot of very long-winded matrix calculations, so we really won’t talk about them here, but there are some nice details written down that show how Green’s resolution can be understood as a sort of “semi-strictification” of twisting cochains.

We also go through the example given in Green’s thesis in a bit more detail, showing how you can calculate the first Chern class of a skyscraper sheaf using all the formalisms that we’ve developed. This method is so nice because you actually end up with explicit Čech-de Rham representatives for the cohomology class, whereas a calculation of the Chern class using, say, short exact sequences and the Whitney sum formula will “only” give you the cohomology class itself.

So what’s left to do? We’ve discussed Green’s construction and shown how it fits into a more general, more abstract framework. We’ve also shown that it agrees with this manual lifting that you can do by equating coefficients of polynomials in cyclic-permutative variables. What more could we hope for?

Well, up until now, everything has been nice, but we haven’t really made things much more useful; we’ve just given everything a much fancier, more longwinded name. It would be much better if we could actually say things in the language of category theory, and, indeed, in the language of (\infty,1)-categories. This would tell us that the construction is somehow “a good one”, in that it respects the inherent structures between all these objects.

It turns out that there are some technical difficulties with even deciding what exactly “a complex of coherent analytic sheaves” should be, but we can get around this somehow, and end up with the following result: there is an equivalence of (\infty,1)-categories between complexes of coherent analytic sheaves and the homotopy colimit (over refinements of covers) of Green complexes (which are exactly these complexes of vector bundles on the nerve arising from Green’s resolution, satisfying this “injective-coface” property, amongst others). As a bonus fact, since X is assumed to be paracompact, for any complex of coherent analytic sheaves, this homotopy colimit will be finite, i.e. we can actually do calculations and write down resolutions. All of what we discussed above amounts to a proof that “simplicial Chern-Weil theory works on Green complexes”, and so this equivalence tells us that it also works on complexes of coherent analytic sheaves, which is exactly what we started out wanting to show!