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 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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