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.

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.'”

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).

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.

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 🙂

Propagation of Sections

Published by brianhepler

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