Propagation of Sections

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 \Omega_0 \subseteq \Omega_1 be two convex (non-empty) open subsets of V.

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Let \mathcal{L}^\bullet 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 \Omega_0 \hookrightarrow \Omega_1 induces an isomorphism in (hyper)cohomology: R\Gamma(\Omega_1;\mathcal{L}^\bullet) \overset{\thicksim}{\to} R\Gamma(\Omega_0; \mathcal{L}^\bullet).  Ok, so what?

So, instead of just say “they have isomorphic cohomology,” you should be thinking “every section of \mathcal{L}^\bullet over \Omega_0 can be extended to all of \Omega_1,” or “sections of \mathcal{L}^\bullet can propagate from \Omega_0 to \Omega_1, in all ‘directions.'”

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The idea of localizing a sheaf \mathcal{F}^\bullet at a point p \in U (say for the moment, U \subseteq \mathbb{R}^n is some open subset) is one of the most basic tools in an algebraic geometer’s tool box.  You forget everything else about how \mathcal{F}^\bullet might behave on U, and just focus on its behavior very close to p.  That is, you’re examining its local behavior at p.  If we want to understand how the cohomology of \mathcal{F}^\bullet might change as we move away from p, we need to examine the microlocal behavior of \mathcal{F}^\bullet at p, in the space of directions from p (really, however, we want ‘codirections’ associated to covectors based at p. That part is coming).

Pick some (non-zero, duh) covector \eta \in T_p^*U, the cotangent space of U to p.  By shrinking U if necessary, we can then assume that \eta is the differential of some smooth function L: U \to \mathbb{R}, and by translation, we can further assume that L(p) = 0.  Although lots of functions might have differential equal to \eta at p, for simplicity, we can just assume that L is some globally defined \mathbb{R}-linear form.  Since, locally, we can safely just picture covectors as ordinary vectors based at p, I like to think of \eta as the “normal vector” to the hyperplane L^{-1}(0) passing through p.  Which makes sense, of course, since we can think of the differential d_p L = \eta here as “the gradient vector \nabla L(p)“, which is always normal to level sets of L. 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).

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The whole point of this is that we want to use this function L to investigate possible changes in the cohomology of \mathcal{F}^\bullet as we move away from p, in the codirection \eta = d_p L.  How would we go about doing that?

Local Morse data!

Think about it.  First, let’s work inside some small open ball W = B_\epsilon^\circ(p) about p.  If every section of \mathcal{F}^\bullet over L^{-1}(-\infty,0) \cap W can be extended passed the “boundary” L^{-1}(0) \cap W, at least a little bit (say to L^{-1}(-\infty,\delta) \cap W for some small \delta > 0), then the cohomology of \mathcal{F}^\bullet over W should be isomorphic to the cohomology over L^{-1}(-\infty,0) \cap W, and this isomorphism should be induced by restriction.

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So, in fancy language, we’d say the relative cohomology complex of (W,W \cap L^{-1}(-\infty,0))  is zero.  Of course, it might happen to be the case that some sections can only propagate across L^{-1}(0) on small neighborhoods of p, so in order to pick up all such sections, we look at the limit \varinjlim_{p \in W \text{open}} \mathbb{H}^k(W , W \cap L^{-1}(-\infty,0); \mathcal{F}^\bullet) for k \in \mathbb{Z}.  A little bit of abstract nonsense then reveals this object is none other than LMD(L,p;\mathcal{F}^\bullet) = R\Gamma_{\{L \geq 0\}}(\mathcal{F}^\bullet)_p, the local Morse data of L at p, with respect to \mathcal{F}^\bullet.

Recalling my last post, this sort of vanishing tells us that the pair (p,\eta) \in T^*U is not in the microsupport of \mathcal{F}^\bullet! Before, I denoted this by \mu supp(\mathcal{F}^\bullet), but from now on, I think I’ll stick with the more popular notation: SS(\mathcal{F}^\bullet) (“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 “\gamma-topology” associated to a closed, convex cone \gamma 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 🙂

Author: brianhepler

I'm a second-year math postdoc at the University of Wisconsin-Madison, and I think math is pretty neat. Especially the more abstract stuff. It's really hard to communicate that love with the general population, but I'm going to do my best.

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