I mentioned last post that one should think of the microsupport of a complex of sheaves in terms of some loose idea of “propagation.” I want to talk about that a little more now. The gist I want you to walk away with is “the microsupport characterizes (co)directions of non-propagation.” Whatever that means.
Let’s start with a simple example. Let V be a (finite dimensional) real vector space, and let be two convex (non-empty) open subsets of V.
Let be a local system on V (i.e., a locally constant sheaf of real vector spaces, considered as a complex concentrated in degree zero). It’s well known that the inclusion of open subsets induces an isomorphism in (hyper)cohomology: . Ok, so what?
So, instead of just say “they have isomorphic cohomology,” you should be thinking “every section of over can be extended to all of ,” or “sections of can propagate from to , in all ‘directions.'”
The idea of localizing a sheaf at a point (say for the moment, is some open subset) is one of the most basic tools in an algebraic geometer’s tool box. You forget everything else about how might behave on , and just focus on its behavior very close to . That is, you’re examining its local behavior at . If we want to understand how the cohomology of might change as we move away from , we need to examine the microlocal behavior of at , in the space of directions from (really, however, we want ‘codirections’ associated to covectors based at . That part is coming).
Pick some (non-zero, duh) covector , the cotangent space of to . By shrinking if necessary, we can then assume that is the differential of some smooth function , and by translation, we can further assume that . Although lots of functions might have differential equal to at , for simplicity, we can just assume that is some globally defined -linear form. Since, locally, we can safely just picture covectors as ordinary vectors based at , I like to think of as the “normal vector” to the hyperplane passing through . Which makes sense, of course, since we can think of the differential here as “the gradient vector “, which is always normal to level sets of . In the general case, you can still make this association, but you’ll need some added structure on your space (e.g., a Riemannian metric).
The whole point of this is that we want to use this function to investigate possible changes in the cohomology of as we move away from , in the codirection . How would we go about doing that?
Local Morse data!
Think about it. First, let’s work inside some small open ball about . If every section of over can be extended passed the “boundary” , at least a little bit (say to for some small ), then the cohomology of over W should be isomorphic to the cohomology over , and this isomorphism should be induced by restriction.
So, in fancy language, we’d say the relative cohomology complex of is zero. Of course, it might happen to be the case that some sections can only propagate across on small neighborhoods of , so in order to pick up all such sections, we look at the limit for . A little bit of abstract nonsense then reveals this object is none other than , the local Morse data of at , with respect to .
Recalling my last post, this sort of vanishing tells us that the pair is not in the microsupport of ! Before, I denoted this by , but from now on, I think I’ll stick with the more popular notation: (“SS” stands for “singular support,” in case you were wondering).
In order to explore further properties of the microsupport of a complex of sheaves, we’ll need a little more machinery at our disposable. In particular, something called the “-topology” associated to a closed, convex cone inside a vector space, and the “non-characteristic deformation lemma,” a technical result at the core of the majority of the ensuing propagation theorems. At the risk of ranting even more, I’ll stop here. I know I haven’t done much new stuff in this post, but I hope I’ve helped elucidate the basic ideas/motivations behind microsupports.
Until next time 🙂