Microsupport and Propagation

So, last time, I briefly mentioned a sheaf-theoretic “Local Morse Datum” for a smooth (Morse) function $f : M \to \mathbb{R}$ at a (non-degenerate) critical point $p \in M$ :

$LMD(f,p) := R\Gamma_{\{f \geq 0\}}(\mathbb{Z}_M^\bullet)_p$

which gives the integral cohomology of the “local Morse datum” of $f$ at $p$ , considered as a pair of spaces. Confusing, I know. It seems like complete overkill at this point, and it is.

So, let’s go deeper.

First, we’ll consider more general objects: the local Morse data of $f$ at $p$ , with respect to the complex of sheaves $\mathcal{A}^\bullet \in D^b(M)$ , denoted

$LMD(f,p; \mathcal{A}^\bullet ) := R \Gamma_{\{ f \geq 0 \}}(\mathcal{A}^\bullet )_p$

Basically, you have the complex of sheaves, $\mathcal{A}^\bullet$ , on $M$ , and you consider sections of this “sheaf” whose support is contained in the subset $\{f \geq 0\} := f^{-1}[0,\infty)$ , and take the stalk cohomology at the point $p \in f^{-1}(0)$ . Sort of like: the sections of the sheaf that propagate in the “positive direction” (where “positive” is taken to be with respect to $f$ (really, the covector $d_pf$ , but we’re not quite there yet 🙂 ) ). If this stalk cohomology vanishes, then the local sections of the sheaf can be “extended” a little bit further away from $p$ , at least in the “positive” direction. That’s supposed to be what I mean by “propagate” here.

Why is this idea useful?

Let $X \subseteq M$ be a sufficiently nice closed subset, so we can give it a Whitney stratification (by which I mean, “satisfies Whitney’s condition (b) at all appropriate times”), $\mathfrak{S}$ . To each stratum $S \in \mathfrak{S}$ , we associate the following subset of the cotangent bundle of $M$ , called the conormal space to S in M:

$T_S^* M := \{ (p,\eta) \in T^*M | p \in S, \eta ( T_p S) = 0 \}$

which consists of covectors in the cotangent bundle which annihilate the various tangent spaces to $S$ , considered as subspaces of the tangent spaces to $M$ . Equivalently (after perhaps endowing $M$ with a Riemannian metric), you can think of these elements $(p,\eta)$ as hyperplanes in $T_pM$ that contain the subspace $T_p S \subseteq T_p M$ .

Let $\overset{\thicksim}{f} : M \to \mathbb{R}$ be a smooth function, $f := \overset{\thicksim}{f}|_X$ its restriction, $\Sigma_\mathfrak{S} f$ the associated stratified critical locus of $f$ . It is any easy exercise to show that, when $\mathfrak{S}$ is a Whitney (a) stratification, $\Sigma_\mathfrak{S} f$ is a closed subset of $X$ . Similarly, the Whitney (a) condition is equivalent to requiring the equality:

$\bigcup_{S \in \mathfrak{S}} T_S^*M = \bigcup_{S \in \mathfrak{S}} \overline{T_S* M}$

Hence, when I say $p \in \Sigma_\mathfrak{S} f$ , there is a unique stratum, say $S$ , for which $p \in \Sigma (f|_S)$ (since the strata are disjoint). And, if you think about it, if $p \in \Sigma (f|_S)$ , we must have

$(p,d_p f) \in T_S^*M$

as $d_p (f|_S) = d_p f(T_p S) = 0$ . Remember this.

Local Stratified Acyclicity (LSA)

Say we have our Whitney stratification, $\mathfrak{S}$ , of $X \subseteq M$ . Then, $\mathfrak{S}$ satisfies something called LSA:

for all $\mathcal{F}^\bullet \in D_\mathfrak{S}^b(X)$ (this means the cohomology sheaves of $\mathcal{F}^\bullet$ are all locally constant with respect to the strata of $\mathfrak{S}$ ), for all (germs of) stratified submersions $f: (M,p) \to (\mathbb{R},0)$ ,

$R \Gamma_{\{ f|_X \geq 0 \} }(\mathcal{F}^\bullet )_p = 0$ .

I’m not going to prove this, but it’s a consequence of a result called the non-characteristic deformation lemma of Kashiwara and Schapira in Sheaves on Manifolds. This tells us that the cohomology sheaves of complexes of sheaves are locally constant if a certain (similar) vanishing condition occurs with germs of submersions.

And now, finally, I can talk about the microsupport of a complex of sheaves, $\mathcal{F}^\bullet \in D^b(M)$ , which encodes the “directions” at given points of $M$ where sections of $\mathcal{F}^\bullet$ “do not propagate.” That is, the directions in which we should expect to detect changes in the cohomology (sheaves) of $\mathcal{F}^\bullet$ . Precisely, the Microsupport of $\mathcal{F}^\bullet$ is the subset $\mu supp(\mathcal{F}^\bullet ) \subseteq T^*M$ , characterized (in the negative…) by:

$(p,\eta) \notin T^*M$ if and only if there exists an open subset $U$ of $(p,\eta)$ in $T^*M$ such that, for all smooth function germs $f : (M,x) \to (\mathbb{R},0)$ with $(x,d_x f) \in U$ , one has $R \Gamma_{\{ f \geq 0 \}}(\mathcal{F}^\bullet )_x = 0$ .

So, if $(p,\eta) \in \mu supp(\mathcal{F}^\bullet)$ , locally, we can say $\eta = d_p f$ for some smooth function germ at $p$ , and if you move in the direction of the “gradient flow” of $f$ in the “positive direction,” $\mathcal{F}^\bullet$ will detect a change in the cohomology of $M$ . I like to picture $f$ to be the germ of some linear form (say in a local coordinate system about $p$ ), and the gradient flow is like looking a family of cross sections of $M$ near $p$ . Moreover, $p$ will be a critical point whatever function we pick, by trivial application of LSA.

I also think this emphasizes the importance of the connection with Morse data: $LMD(f,p; \mathcal{F}^\bullet ) \neq 0$ if and only if $(p,d_p f) \in \mu supp(\mathcal{F}^\bullet)$ .

Now, say we’ve got our sufficiently nice closed subset $X \subseteq M$ , with Whitney stratification $\mathfrak{S}$ , equipped with inclusion map $i: X \hookrightarrow M$ . Say we’ve got some smooth function $f: M \to \mathbb{R}$ . We know that, for $p \in S \in \mathfrak{S}$ , $p \in \Sigma (f|_S)$ if and only if $(p, d_p f) \in T_S^*M$ . Then, via the isomorphism

$R \Gamma_{\{ f|_X \geq 0 \}}(\mathcal{F}^\bullet )_p \cong R \Gamma_{\{ f \geq 0 \}} (Ri_* \mathcal{F}^\bullet )_p$

and an application of LSA, this quantity vanishes if and only if $f|X$ is a stratified submersion. Consequently, we see

$\mu supp(Ri_* \mathcal{F}^\bullet) \subset \bigcup_{S \in \mathfrak{S}} T_S^*M$

Now…what does the microsupport reveal about the local Morse data of functions with stratified critical points? That’s where things will start to get interesting. Until next time.

References:

M. Kashiwara and P. Schapira; Sheaves on Manifolds.

J. Schurmann; Topology of Singular Spaces and Constructible Sheaves.

Microsupport and Propagation

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