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.


Literature, Originality, Influence, and the Anxiety Thereof

I really do plan to get back to some math soon. I thought I’d share an argument that I first learned from the essay “The Literature of Exhaustion” by John Barth. It is something that used to come up all the time when I was an undergrad music major. It usually comes up now in the form of literature. I’ll phrase it in terms of literature, since this is the form it appears in Barth’s essay, but it works for any art form.

The reason this came up recently is because I was watching an interview with Patrick Rothfuss. From what I’ve seen, he takes the craft of writing about as seriously as any author I’ve seen, and he really wants to better himself as an author in any way possible. He was asked what sorts of fiction he reads outside of the sci-fi/fantasy genre. I was shocked to hear that he basically doesn’t. It reminded me of this argument. Similarly, the music composition students that I used to talk with had basically no interest in listening to or analyzing current living composers.

I can’t remember now, but I think I’ve made this argument on the blog somewhere before. It’s well worth repeating, since people always seem surprised by it when it comes up. There is this disconnect that because art is “subjective” it doesn’t build on itself. People seem to have the opinion that art just spreads off in random branches of originality and you don’t have to pay attention to what your contemporaries are doing. Some go so far as to claim that paying attention to your contemporaries blocks your ability to be truly original through subconscious influence.

Here is part of Barth’s argument. No one would ever dare to say this type of thing about any branch of science or math. You would be laughed at. Imagine saying you could do something completely original in physics by ignoring the last 50 years of research so you aren’t influenced. The obvious problem with this is that in order to do something new, you will have to completely reinvent all of the past 50 years of physics (at least in your area) before getting to the new part. What is the point of trying to do that when you could just intentionally learn it in a small fraction of the time and get on to your new ideas. In fact, even coming up with a new idea might be impossible without having seen the recent advances that open your mind to ideas that were inconceivable beforehand.

To put it bluntly, trying to make some awesome original art without being up-to-date on what has been done doesn’t make you a visionary. It makes you an idiot. The main objection to this is probably that in art as opposed to science you don’t have the same type of building. You don’t need to be completely current on what every contemporary author is doing in order to build off in some direction or try something new. This is in part true, but let’s try to put this in perspective.

If you were to take a one semester course whose primary goal was to expose you to as many significant advances in just some very narrow frame like American literature of the past 40 years it couldn’t be done thoroughly. This is with an intensive study by someone who knows what they are doing with this goal in mind. Think about how hopeless it would be to try to invent all these ideas yourself whenever you need one of them. It is just about as hopeless as the scientist who tries to ignore the past 40 years of science.

I claim that not only is it a good idea for a serious genre author to make an attempt to keep up with modern literature, but it is almost certainly the most important thing they can do to better themselves. Forget about the worries of being unoriginal due to influence. You are certain to be unoriginal if you don’t keep up, whereas if you know what people have been doing, then at least you have some chance of building upon it in a unique direction. It will not only improve your writing to have these modern techniques at your disposal, but it will make you a much more interesting writer as well. I quote Barth, “In any case, to be technically out of date is likely to be a genuine defect: Beethoven’s Sixth Symphony or the Chartres cathedral, if executed today, might be simply embarrassing ….”

Nothing is as frustrating as getting into these types of arguments with people who want to think that being a good artist is all about this lackadaisical, touchy-feely, everything goes attitude. That is just some romantic fantasy. All the great artists have put a lot of hard work and study into it, and part of that study is understanding what other great people in your craft have done. Don’t take my word for it. Try it out. You’ll probably find that not having to reinvent the wheel every time you need a particular technique actually frees up your energy to use on creating something that is actually new.

Examples: For anyone interested in these sorts of ideas you should check out The Friday Book by John Barth which has his essay in it. The ultimate example I’d have to say is Harold Bloom’s The Anxiety of Influence which is basically a book long case study of how some of the great poets influenced each other and overcame those influences to create something new (an impossible task if they weren’t reading each other I might point out).

I edited this post since it was too long already. I had included three interesting examples originally, but realized it almost hurts the argument to see examples. All the examples I thought of could be written off as singular, fringe cases, so I haven’t included them now. As a game, just take any of the hundreds of lists out there with names like “100 Best Novels” or whatever and try to find even one novel on that list that didn’t liberally borrow techniques from a contemporary. For you genre writers out there that think you can get away with staying within the genre, a quick glance at the Modern Library list and the Time Magazine list shows many great genre authors like George Orwell, Robert Heinlein, Kurt Vonnegut, J.R.R. Tolkien, William Gibson, Neal Stephenson, Samuel Delany, Philip K. Dick, and more. Every one of these authors would have been severely hindered without being up to date on modern fictional techniques that mostly weren’t appearing in the genre.