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.

Author: brianhepler

I'm a third-year math postdoc at the University of Wisconsin-Madison, where I work as a member of the geometry and topology research group. Generally speaking, I think math is pretty neat; and, if you give me the chance, I'll talk your ear off. 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 to show you a world of pure imagination.