Classical Local Systems

I lied to you a little. I may not get into the arithmetic stuff quite yet. I’m going to talk about some “classical” things in modern language. In the things I’ve been reading lately, these ideas seem to be implicit in everything said. I can’t find this explained thoroughly anywhere. Eventually I want to understand how monodromy relates to bad reduction in the {p}-adic setting. So we’ll start today with the different viewpoints of a local system in the classical sense that are constantly switched between without ever being explained.

You may need to briefly recall the old posts on connections. The goal for the day is to relate the three equivalent notions of a local system, a vector bundle plus flat connection on it, and a representation of the fundamental group. There may be some inaccuracies in this post, because I can’t really find this written anywhere and I don’t fully understand it (that’s why I’m making this post!).

Since I said we’d work in the “classical” setting, let’s just suppose we have a nice smooth variety over the complex numbers, {X}. In this sense, we can actually think about it as a smooth manifold, or complex analytic space. If you want, you can have the picture of a Riemann surface in your head, since the next post will reduce us to that situation.

Suppose we have a vector bundle on {X}, say {E}, together with a connection {\nabla : E\rightarrow E\otimes \Omega^1}. We’ll fix a basepoint {p\in X} that will always secretly be lurking in the background. Let’s try to relate this this connection to a representation of the fundamental group. Well, if we look at some old posts we’ll recall that a choice of connection is exactly the same data as telling you “parallel transport”. So what this means is that if I have some path on {X} it tells me how a vector in the fiber of the vector bundle moves from the starting point to the ending point.

Remember, that we fixed some basepoint {p} already. So if I take some loop based at {p} say {\sigma}, then a vector {V\in E_p} can be transported around that loop to give me another vector {\sigma(V)\in E_p}. If my vector bundle is rank {n}, then {E_p} is just an {n}-dimensional vector space and I’ve now told you an action of the loop space based at {p} on this vector space.

Visualization of a vector being transported around a loop on a torus (yes, I’m horrible at graphics, and I couldn’t even figure out how to label the other vector at p as \sigma (V)):

This doesn’t quite give me a representation of the fundamental group (based at {p}), since we can’t pass to the quotient, i.e. the transport of the vector around a loop that is homotopic to {0} might be non-trivial. We are saved if we started with a flat connection. It can be checked that the flatness assumption gives a trivial action around nullhomotopic loops. Thus the parallel transport only depends on homotopy classes of loops, and we get a group homomorphism {\pi_1(X, p)\rightarrow GL_n(E_p)}.

Modulo a few details, the above process can essentially be reversed, and hence given a representation you can produce a unique pair {(E,\nabla)}, a vector bundle plus flat connection associated to it. This relates the latter two ideas I started with. The one that gave me the most trouble was how local systems fit into the picture. A local system is just a locally constant sheaf of {n}-dimensional vector spaces. At first it didn’t seem likely that the data of a local system should be equivalent to these other two things, since the sheaf is locally constant. This seems like no data at all to work with rather than an entire vector bundle plus flat connection.

Here is why algebraically there is good motivation to believe this. Recall that one can think of a connection as essentially a generalization of a derivative. It is just something that satisfies the Leibniz rule on sections. Recall that we call a section, {s}, horizontal for the connection if {\nabla (s)=0}. But if this is the derivative, this just means that the section should be constant. In this analogy, we see that if we pick a vector bundle plus flat connection, we can form a local system, namely the horizontal sections (which are the locally constant functions). If you want an exercise to see that the analogy is actually a special case, take the vector bundle to be the globally trivial line bundle {\mathcal{O}_X} and the connection to be the honest exterior derivative {d:\mathcal{O}_X\rightarrow \Omega^1}.

The process can be reversed again, and given any locally constant sheaf of vector spaces, you can cook up a vector bundle and flat connection whose horizontal sections are precisely the sections of the sheaf. Thus our three seemingly different notions are actually all equivalent. I should point out that part of my oversight on the local system side was thinking that a locally constant sheaf somehow doesn’t contain much information. Recall that it is still a sheaf, so we can be associating lots of information on large open sets and we still have restriction homomorphisms giving data as well. Next time we’ll talk about some classical theorems in differential equation theory that are most easily proved and stated in this framework.

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Stratification 2

Before defining stratification, we’ll look at what this notion is in what is hopefully a more familiar context. Let’s forget about all the PD stuff for today (but keep it in mind for later). Suppose {S} is a scheme and {X} is smooth of finite type over {S}. The diagonal is a closed immersion {\Delta: X\rightarrow X\times_S X}. Suppose it is defined by the quasi-coherent ideal sheaf {\mathcal{I}} (it is generated by things of the form {t\otimes 1 - 1 \otimes t}). So far all this should feel very familiar from our setup in the last post.

Rather than worry about PD things, we’ll define the {n}-th infinitesimal neighborhood of {\Delta} to be {X^{(n)}}, the subscheme defined by {\mathcal{I}^{n+1}}. There are natural inclusions {X\hookrightarrow X^{(2)}\hookrightarrow X^{(3)}\rightarrow \cdots}. and all of these sit inside {X\times X}. Define {p_1} and {p_2} to be the first and second projections {X\times X\rightarrow X}. Given a quasi-coherent sheaf {\mathcal{E}}, we already have a notion of connection. Recall that it is just linear map {\nabla: \mathcal{E}\rightarrow \mathcal{E}\otimes \Omega^1} that satsifies the Leibniz rule.

Grothendieck defined a connection on {\mathcal{E}} to be an isomorphism {(p_1^*\mathcal{E})|_{X^{(2)}}\stackrel{\sim}{\rightarrow} (p_2^*\mathcal{E})|_{X^{(2)}}} which restricts to the identity on {\Delta}. I’ll give you the punchline up front. This definition is equivalent to the other one! If you look at the last post, it should be clear that this is the definition that will extend easier in the PD case. Let’s try to understand what this definition is saying.

This definition is somehow related to parallel transport. How could we think about parallel transport? Suppose you have some pointed {S}-scheme, say {T} with {t}. Consider a closed immersion {f:(T,t)\rightarrow (X, x)}. Maybe this is like a path and you remember the starting point (but it doesn’t have to be). There is always the constant map {f_x: (T,t)\rightarrow (X,x)} which just sends all of {T} to {x}. Parallel transport along {f(T)} from {x} of {\mathcal{E}} should be an isomorphism {\mathcal{E}|_{f(T)}\stackrel{\sim}{\rightarrow} \mathcal{E}|_{f_x(T)}} which restricts to be the identity at {x}.

Strictly speaking what I just wrote is nonsense, so what do I mean? If we restrict {\mathcal{E}} to the image of {T} I want to be able to choose a trivialization to give a linear isomorphism with the trivial bundle having fiber {\mathcal{E}|_{x}}. If you are thinking in terms of vector bundles, you could also think of it as a compatible choice of isomorphisms of the fiber at all points of {F(T)} with the fiber at {x} and the isomorphism at {x} must be the identity.

Now the definition of connection we gave should probably more accurately be described as “first-order” parallel transport where for us we should be thinking that {T=\mathrm{Spec}(k[\epsilon])}. Let’s check that our new notion of connection gives parallel transport with this {T}. Choose a {k}-point on {x\in\Delta}, then {f: T\rightarrow X\times X} is just going to {x} and choosing a tangent vector there. Since we have by definition an isomorphism {p_1^*\mathcal{E}|_{X^{(2)}}\rightarrow p_2^*\mathcal{E}|_{X^{(2)}}} we can just restrict this further to {f(T)} which lies in {X^{(2)}} to get the parallel transport isomorphisms. This shows that connections give parallel transport along tangent vectors (i.e. along first-order infinitesimally short paths).

Now nothing is stopping us from continuing in this fashion. What happens if we take {X_3^{(2)}} as the first-order infinitesimal neighborhood of the diagonal in {X\times X\times X}. Then we have three projections {X\times X\times X\rightarrow X\times X} which we’ll call {p_{i,j}} for projecting onto the {i} and {j} factors. Using the definition of connection we can now obtain {\epsilon_{i,j}: p_i^*\mathcal{E}|_{X_3^{(2)}}\stackrel{\sim}{\rightarrow} p_j^*\mathcal{E}|_{X_3^{(2)}}} which just comes from first pulling back using {p_{i,j}} then restricting.

Let’s think back to when we talked about connections months ago. The next major definition was what it meant to be integrable. We defined the curvature {K(\nabla)} to be the composition {\mathcal{E}\rightarrow \mathcal{E}\otimes \Omega^1\rightarrow \mathcal{E}\otimes \Omega^2} and {\nabla} was integrable if the curvature was {0}. Or in other words, if the associated sequence was actually a complex.

Using our new definition we get that a connection is integrable if {\epsilon_{1,3}=\epsilon_{2,3}\circ \epsilon_{1,2}}. In other words, if the isomorphisms coming from the associated {X\times X\times X} satisfy some sort of cocycle condition. In characteristic {0} being integrable actually guarantees that you can lift the isomorphisms to all {n}-th order neighborhoods. This gives {n}-th order parallel transport, meaning we get parallel transport using {T=\mathrm{Spec}(k[x]/(x^n))}.

Now everything is compatible here (we’re lifting the isomorphisms, meaning they restrict to the previous one). Thus we actually have directed systems to take a limit. This gives us what could maybe be called formal local parallel transport. Believe it or not, this is exactly the type of thing we are after in the PD case. It is basically the purpose of building a notion of “{x^n/n!}” so that we can get some sort of formal power series. I think that is a good enough reminder of this “classical” case. Maybe if you are super motivated you can go to the previous post and work out how these definitions extend. The setup is all there.

Gauss-Manin Connection 2

It’s probably been awhile since you read the first post in this series, so I’ll quickly remind you of the key point. {S} is a smooth scheme over a field {k}. We fixed connection {\rho} on {\mathcal{E}}. Then given a derivation {\delta} corresponding to {D: \Omega\rightarrow \mathcal{O}_S}, then for any element of {\mathcal{D}er_k(\mathcal{O}_S)}, the sheaf of germs of {k}-derivations we can compose the maps we have and we get {\overline{D}\in \mathcal{E}nd_k(\mathcal{E})}.
So every connection gives a map {\mathcal{D}er_k(\mathcal{O}_S)\rightarrow \mathcal{E}nd_k(\mathcal{E})}, {\delta\mapsto \overline{D}} and we had a relation between {D} and {\overline{D}}, and any such map satisfying the relation comes from a connection.

Now we’ll go back to the construction of the Gauss-Manin connection from last time. We haven’t actually checked that it is a connection or that it is flat. Recall that it is just one of the maps we get from the spectral sequence associated to the filtration of the complex {\Omega_{X/k}^\cdot}. Now the filtration is compatible with taking wedge products ({F^i\wedge F^j\subset F^{i+j}}) and the functors {\mathbf{R}^q\pi_*} are multiplicative, so we have a product structure on the terms of the spectral sequence as follows.

If we take sections of the sheaves over an open, then {E^{p,q}_r\times E^{p',q'}_r\rightarrow E^{p+p', q+q'}_r} by {(e,e')\mapsto e\cdot e'}. If you want the actual construction see Godement. The product satisfies a few important properties. We have a type of anti-commutativity {e\cdot e'=(-1)^{(p+q)(p'+q')}e'\cdot e}. Also we know how it behaves under the differential map: {d_r(e\cdot e')=d_r(e)\cdot e'+ (-1)^{p+q}e\cdot d_r(e')}.

In particular, let’s look at what this product rule for the differential is for the Gauss-Manin map. For {\nabla=d_1^{0,q}:E_1^{0, q}\rightarrow E_1^{1,q}} which is really mapping {\mathcal{H}^q_{dR}(X/S)\rightarrow \Omega_{S/k}\otimes \mathcal{H}^q_{dR}(X/S)}, the differential is really just {d_{S/k}\otimes Id}. Thus that rule says that {\nabla(\omega\cdot e)=d\omega\cdot e+(-1)^0\omega\cdot \nabla(e)}. So it is a connection!

The curvature is easily seen to be {d_1^{1,q}\circ d_1^{0,q}} and since the {d_1}‘s are maps of a complex we get that it is {0}, and hence {\nabla} is flat and hence the Gauss-Manin connection is integrable. We’ve now proved the theorem that any smooth {k}-morphism of smooth {k}-schemes gives rise to a canonical integrable connection on the relative de Rham cohomology sheaves that is compatible with the cup product.

If you want a more explicit way to see what the map is see the paper, but it is kind of tedious since writing out how it appears in the spectral sequence you will quickly find that it is the connecting homomorphism when taking the long exact sequence after applying the functor {\mathbf{R}^q\pi_*} to the exact sequence {0\rightarrow \mathrm{gr}^{p+1}\rightarrow F^p/F^{p+2}\rightarrow \mathrm{gr}^p\rightarrow 0}.

This is the third post on this topic, and I haven’t given you a reason to care yet. Here’s why we should care. One would hope (via a conjecture of Grothendieck) that there is some sort of relative de Rham Leray Spectral Sequence: {E_2^{p,q}=\mathbf{H}^p(S, \Omega_{S/k}^\cdot \otimes_{\mathcal{O}_S} \mathcal{H}^q(X/S))\Rightarrow H_{dR}^{p+q}(X/k)}. For the {E_2}-term to make any sort of sense we needed {\Omega_{S/k}^\cdot \otimes_{\mathcal{O}_S} \mathcal{H}^q(X/S)} to be a complex, and since the Gauss-Manin connection is integrable it is a complex. Also, {H_{dR}^{p+q}(X/k)} is defined to be {\mathbf{H}^{p+q}(X, \Omega_{X/k}^\cdot)}.

It turns out that when {S} is affine such a spectral sequence exists. In case you’re wondering, affineness is needed for a nice proof of this because it makes certain cohomologies vanish. Deligne has proved it in a more complicated way when {S} is not affine (but still with our standing assumptions). This is of great importance in proving every K3 surface lifts from characteristic {p} to characteristic {0}.

Connections and Curvature … on Schemes! … in Characteristic p?!

I feel bad about my absence. I lasted posted during winter break, and now winter quarter is completely over. I kept meaning to do a series on “well-known” algebraic geometry results and constructions that don’t appear with any amount of thoroughness in the references. I thought it would be good to get that information out there. Unfortunately, I had already written these things down into a notebook and just couldn’t motivate myself to type something up that I already had. Anyway, one thing led to another and I didn’t do any posts. I’m not sure why I’m trying to justify my absence with an excuse.

Recently I’ve been typing up a translation of Deligne’s argument (written down by Illusie) that every K3 surface in characteristic {p>0} lifts to characteristic {0}. I’m not to the point of trying to understand it, but I wanted a typed version, so that when I get the background material (namely crystalline cohomology!) and go to understand it, I can just fill in the details into my typed notes quickly and easily. I also was curioius as to the overall format of the argument.

This led me to the 1968 paper by Katz and Oda called On the Differentiation of de Rham Cohomology Classes with Respect to Parameters. The next few posts will be about the main result from this paper. It is really quite amazing.

First, some definitions. We’ll always be working with a smooth scheme {S} over a field {k} (no assumptions here!). Let {\mathcal{E}} be a quasi-coherent sheaf of {\mathcal{O}_S}-modules. We’ll write {\Omega} for {\Omega^1_{S/k}} and unless otherwise noted, all tensor products will be over {\mathcal{O}_S}. We say that {\nabla} is a connection on {\mathcal{E}} if it is a homomorphism {\nabla: \mathcal{E}\rightarrow \Omega\otimes \mathcal{E}} that satisfies the “Leibniz rule”.

In other words, {\nabla(fg)=f\nabla(g)+df\otimes g}. This is the standard shorthand meaning {\nabla(U): \mathcal{E}(U)\rightarrow \Omega(U)\otimes \mathcal{E}(U)} satisfies the rule where {f\in \mathcal{O}_S(U)}, {g\in \mathcal{E}(U)} and {df} is the image of {f} under the universal map {\mathcal{O}_S\rightarrow \Omega}.

Given a connection {\rho}, we get homomorphisms for all {i}, {\rho_i: \Omega^i\otimes\mathcal{E}\rightarrow \Omega^{i+1}\otimes \mathcal{E}}. These are given by {\rho_i(\omega\otimes e)=d\omega \otimes e+(-1)^i \omega\wedge \rho(e)}.

The notation is just the one that makes sense: {\rho(e)\in \Omega\otimes \mathcal{E}}, so it looks like {\tau\otimes \epsilon}. So we define {\omega\wedge \rho(e)=\omega\wedge(\tau\otimes \epsilon)} to be {(\omega\wedge \tau)\otimes \epsilon\in \Omega^{i+1}\otimes \mathcal{E}}.

Now we define the curvature of the connection {K:\mathcal{E}\rightarrow \Omega^2\otimes \mathcal{E}} to be the map {\rho_1\circ \rho}. The curvature is related to the other {\rho_i} by an easy check {\rho_{i+1}\circ \rho_i(\omega\otimes e)=\omega\wedge K(e)}.

This gives some sort of meaning to the curvature now. If the curvature is {0}, then the natural de Rham-like sequence we get from a connection by stringing together the {\rho_i} as follows {0\rightarrow \mathcal{E}\stackrel{\rho}{\rightarrow} \Omega\otimes\mathcal{E}\stackrel{\rho_1}{\rightarrow} \Omega^2\otimes \mathcal{E}\rightarrow \cdots} is an honest complex that we can take cohomology with respect to, since {\rho_{i+1}\circ \rho_i=0}.

When this happens we call the connection {\rho} integrable. Now let {\mathcal{D}er_k(\mathcal{O}_S)} be the sheaf of germs of {k}-derivations of {\mathcal{O}_S} into itself. From the fact that the module of differentials is a representing object, we get that as a sheaf of {\mathcal{O}_S}-modules, {\mathcal{D}er_k(\mathcal{O}_S)\simeq \mathcal{H}om_{\mathcal{O}_S}(\Omega, \mathcal{O}_S)}.

Let {\mathcal{E}nd_k(\mathcal{E})} be the sheaf of germs of {k}-linear endomorphisms of {\mathcal{E}}. Given any connection {\rho} on {\mathcal{E}} we get an induced {\mathcal{O}_S}-linear map {\mathcal{D}er_k(\mathcal{O}_S)\rightarrow \mathcal{E}nd_k(\mathcal{E})} as follows. Let {\delta} be a derivation, then it corresponds to a map {D: \Omega\rightarrow \mathcal{O}_S}.

So consider the composition {\overline{D}:\mathcal{E}\rightarrow \Omega\otimes \mathcal{E}\rightarrow \mathcal{O}_S\otimes \mathcal{E}\simeq \mathcal{E}}, where the first is the connection and the second is {D\otimes Id}. This gives the map {\mathcal{D}er_k(\mathcal{O}_S)\rightarrow \mathcal{E}nd_k(\mathcal{E})} as {\delta\mapsto \overline{D}}.

Lastly for today, note that we get a nice relation between {D} and {\overline{D}} as follows {\overline{D}(fe)=D(f)e+f\overline{D}(e)} and that any map {\mathcal{D}er_k(\mathcal{O}_S)\rightarrow \mathcal{E}nd_k(\mathcal{E})} satisfying this relation comes from a unique connection on {\mathcal{E}}.

Today was just a bunch of notation and definitions, but next time it should get more interesting.