# 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:

- coherent sheaves often do
*not*admit global resolutions by locally-free sheaves; - 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:

- figure out how to do Chern-Weil theory for vector bundles, i.e. look
at
*simplicial connections*; - 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);
- 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!