# Working Microlocally

So, by now, we have some ideas about what these object $SS(\mathcal{F}^\bullet) \subset T^*X$ are, as (co)directions of non-propagation of sections of the various cohomology sheaves of the complex $\mathcal{F}^\bullet$.  But that’s really only scratching the surface, as all we’ve done so far is fiddle around with the definition of microsupport.  For a general complex of sheaves $\mathcal{F}^\bullet \in D^b(X)$ (X a real, smooth manifold), there isn’t much hope for mere mortals like ourselves in calculating $SS(\mathcal{F}^\bullet)$.  But, things get a little brighter when we restrict to those objects whose cohomology sheaves are all constructible; for $\mathcal{F}^\bullet \in D_c^b(X)$, there exists a stratification $\mathfrak{S}$ of $X$ by smooth submanifolds such that

$SS(\mathcal{F}^\bullet) \subseteq \bigcup_{S \in \mathfrak{S}} \overline{T_S^* X}$.

Specifically, in addition to being $\mathbb{R}^+$-conic and closed, $SS(\mathcal{F}^\bullet)$ is also a real subanalytic, isotropic subset of $T^*X$ (with respect to the usual symplectic structure on the cotangent bundle).

Example:

One of the easiest examples of this is when we consider the constant sheaf on X.  Suppose we’re using $\mathbb{Z}$ as our base ring for $D^b(X)$, and let $N \subseteq X$ be a closed submanifold.  To $N$, we can associate the sheaf $\mathbb{Z}_N^\bullet \in D_c^b(X)$ (considered as a complex of sheaves concentrated in degree zero), whose stalks are $(\mathbb{Z}_N^\bullet)_x \cong \mathbb{Z}$ if $x \in N$, and $\cong 0$ if $x \notin N$.  Then, it’s not too hard to show the equality $SS(\mathbb{Z}_N^\bullet) = T_N^*X$, the full conormal bundle to $N$ in $X$.

Another good toy example is when $X = \mathbb{R}$, and we’re considering the constant sheaf on a closed interval $[a,b] \subset \mathbb{R}$, extended by zero to all of $\mathbb{R}$.  What would this sheaf’s microsupport be?

Since almost all the objects we’ll be interested in are in $D_c^b(X)$, we should explore the microsupport of these complexes as deeply as we can, from as many perspectives as we can find (morally, the “Yoneda lemma” approach (http://mathoverflow.net/a/3223)).

The perspective I want to talk about today is from the (triangulated) derived categories $D^b(X;\Omega)$, from Kashiwara and Schapira’s Sheaves on Manifolds ([K-S]), that are associated to subsets $\Omega \subseteq T^*X$.  Briefly, $D^b(X;\Omega)$ is the localization of $D^b(X)$ by the full subcategory of objects whose microsupport is disjoint from $\Omega$.  Then, we want to say that working “microlocally” with some complex of sheaves $\mathcal{A}^\bullet$ on $X$ means to consider $\mathcal{A}^\bullet$ as an object of the category $D^b(X;\Omega)$ for some subset $\Omega$.  This should seem pretty reasonable; If $SS(\mathcal{A}^\bullet) \cap \Omega = \emptyset$, then $\mathcal{A}^\bullet$ has no interesting microlocal behavior on $\Omega$ and can’t really interact in a meaningful way with other, non-zero, objects of $D^b(X;\Omega)$.

So, now it’s time for the actual details.  Let $V$ be a subset of $T^*X$.  We define $D_V^b(X)$ to be the full subcategory of $D^b(X)$ consisting of those objects $\mathcal{B}^\bullet$ such that $SS(\mathcal{B}^\bullet) \subseteq V$.  Additionally, this is a triangulated category, with distinguished triangles (naturally) those of $D^b(X)$ whose objects belong to $D_V^b(X)$.  Then, $Ob(D_V^b(X))$ is a null-system in $D^b(X)$, and we may define the relevant localization.  That is, we set $D^b(X;\Omega) := D^b(X)/Ob(D_V^b(X))$ (where $\Omega := T^*X \backslash V$).  In the case we’ll usually use, $\Omega = \{\omega\}$ for some covector $\omega \in T^*X$, we’ll be lazy and write $D^b(X;\omega)$ for the localized category.

One of the first things we can say about $D^b(X;\Omega)$ is: a morphism $\mathcal{A}^\bullet \overset{u}{\to} \mathcal{B}^\bullet$ in $D^b(X)$ becomes an isomorphism in $D^b(X;\Omega)$ (or, we say, “on $\Omega$“) if we can embed $u$ in a distinguished triangle $latex \mathcal{A}^\bullet \overset{u}{\to} \mathcal{B}^\bullet \to \mathcal{C}^\bullet \overset{+1}{\to}$ with $SS(\mathcal{C}^\bullet) \cap \Omega = \emptyset$.

Let’s play with some examples, and see how working in $D^b(X;\Omega)$ differs from working in $D^b(X)$.

Example:

Let $X = \mathbb{R}$.  To the closed subset $\{x | x \geq 0\}$, there is a distinguished triangle in $D^b(\mathbb{R})$:

$\mathbb{Z}_{\{x | x < 0\}}^\bullet \to \mathbb{Z}_\mathbb{R}^\bullet \to \mathbb{Z}_{\{x | x \geq 0\}}^\bullet \overset{+1}{\to}$

from this triangle, it follows that the complexes $\cdots 0 \to \mathbb{Z}_{\{x | x < 0\}} \to \mathbb{Z}_\mathbb{R} \to 0 \cdots$ and $\mathbb{Z}_{\{x | x \geq 0\}}^\bullet [-1]$ are isomorphic in $D^b(\mathbb{R})$ (i.e., there is a morphism between the complexes that induces an isomorphism on cohomology).

Let’s now consider the same distinguished triangle, but this time in the category $D^b(\mathbb{R};\Omega)$, where $\Omega = \{ (x,\xi) \in T^*\mathbb{R} | \xi > 0 \}$.  After a quick calculation, we obtain:

• $SS(\mathbb{Z}_\mathbb{R}^\bullet) = \mathbb{R} \times \{0\}$,
• $SS(\mathbb{Z}_{\{x | x \geq 0\}}^\bullet) = \{(0,\xi) | \xi \geq 0\} \cup \{(x,0)| x > 0 \}$, and
• $SS(\mathbb{Z}_{\{x | x < 0 \}}^\bullet) = \{(0,\lambda) | \lambda \leq 0 \} \cup \{(x,0) | x < 0\}$.

whence it follows that $\mathbb{Z}_{\{x | x \geq 0 \}}^\bullet \cong \mathbb{Z}_{\{x | x < 0\}}^\bullet [1]$ in $D^b(\mathbb{R}; \Omega)$, as well as $\mathbb{Z}_{\{0\}}^\bullet \cong \mathbb{Z}_{\{x | x \geq 0\}}^\bullet$ in $D^b(\mathbb{R}; \Omega)$.

Why should you care?

Okay, that’s cute.  But why are these categories useful?  Why should you bother to play with $D^b(X;\Omega)$ at all?

Earlier in this post, I mentioned that, for $\mathcal{F}^\bullet$ constructible, $SS(\mathcal{F}^\bullet)$ is an $\mathbb{R}^+$-conic, subanalytic, isotropic subset of $T^*X$.  Similarly, using $D^b(X;\Omega)$, I want to give a “microlocal” characterization of those $\mathcal{F}^\bullet \in D^b(X)$ whose microsupport is contained in an involutive submanifold of $T^*X$.  Particularly, when the microsupport is contained in the conormal bundle $T_Y^*X$ to some submanifold $Y$ of $X$ (as it turns out, the microsupport is ALWAYS an involutive subset of the cotangent bundle).  Hopefully, this pursuit will tell us something new and interesting about the objects of our favorite category, $D_c^b(X)$.   Eventually (not this time!) we’ll put these two things together, and see what happens along a Lagrangian submanifold; this leads to Kashiwara and Schapira’s notion of pure sheaves.  These are pretty neat.

Proposition 1: ([6.6.1, K-S])

Suppose $Y$ is a closed submanifold of $X$, and let $f: Y \hookrightarrow X$ denote the associated smooth, closed embedding.  Let $p \in T_Y^*X$ (non-zero, or else this is a stupid exercise), and let $\mathcal{F}^\bullet \in D^b(X)$.  Let $\pi : T^*X \to X$ denote the projection of the cotangent bundle.   Assume further that, in a neighborhood of $p$,  $SS(\mathcal{F}^\bullet) \subseteq \pi^{-1}(Y)$.  Then, there exists a $\mathcal{G}^\bullet \in D^b(Y)$ such that $\mathcal{F}^\bullet \cong f_* \mathcal{G}^\bullet$ in $D^b(X;p)$.

(Quick sketch: Basically, we want to successively, microlocally, cut up $SS(\mathcal{F}^\bullet)$ by restricting to various subsets of $X$, and throw away those pieces whose microsupport doesn’t meet $p$.)

Detailed Sketch:

By induction on the codimension of $Y$ in $X$, we may assume $Y$ is a hypersurface, say given by $Y = \{\varphi = \}$, with $p = (x_0, d_{x_0} \varphi)$.  Set $\Omega^{\pm} = \{x \in X | \pm \varphi(x) > 0 \}$, so we have the open embeddings $j_{\pm} : \Omega^\pm \hookrightarrow X$. By [6.3.1, K-S], there is an inclusion

$SS(Rj_{-*}j_{-}^*\mathcal{F}^\bullet) \subseteq SS(j_{-}^*\mathcal{F}^\bullet ) \hat{+} N^*(\Omega^-)$,

where (in our case here) $N^*(\Omega^-) \subseteq T^*X$ consists of the zero section of $T^*X$, together with those covectors $(y,\lambda d_y \varphi) \in T_Y^*X$ where $\lambda < 0$.  Vaguely, the $\hat{+}$ operation characterizes those covectors arising as limits of sequences of covectors in $SS(j_{-}^*\mathcal{F}^\bullet) + N^*(\Omega^-)$ (see [6.2.3, K-S]).

Since $p$ is not in the right hand side, we must have $p \notin SS(Rj_{-*}j_{-}^*\mathcal{F}^\bullet)$.  Hence, from the adjunction distinguished triangle

$R\Gamma_{\{\varphi \geq 0\}} \mathcal{F}^\bullet \to \mathcal{F}^\bullet \to Rj_{-*}j_{-}^*\mathcal{F}^\bullet \overset{+1}{\to}$

in $D^b(X)$, we see that $R\Gamma_{\{\varphi \geq 0\}}\mathcal{F}^\bullet \cong \mathcal{F}^\bullet$ in $D^b(X;p)$.  So, we might as well have assumed from the start that $supp(\mathcal{F}^\bullet) \subseteq \{\varphi \geq 0\}$.  Similarly, we find (by [6.3.1, K-S]) that $p \notin SS(Rj_{+!}j_{+}^*\mathcal{F}^\bullet)$, and see that $\mathcal{F}^\bullet \cong \mathcal{F}_Y^\bullet = f_*f^*\mathcal{F}^\bullet$ in $D^b(X;p)$, using the distinguished triangle

$Rj_{+!}j_+^*\mathcal{F}^\bullet \to \mathcal{F}^\bullet \to i_{-*}i_-^*\mathcal{F}^\bullet \overset{+1}{\to}$

(where $i_- : \{ \varphi \leq 0 \} \hookrightarrow X$ is the closed complement of $\Omega^+$ in $X$), and using the fact that $supp(\mathcal{F}^\bullet \subseteq \{\varphi \geq 0\}$.  All together, this gives us the desired isomorphism $\mathcal{F}^\bullet \cong f_*f^*\mathcal{F}^\bullet$ in $D^b(X;p)$.

Q.E.D.

Proposition 2:  [6.6.1, K-S]

Same as the set up of proposition 1, but now we assume that $SS(\mathcal{F}^\bullet) \subseteq T_Y^*X$ in a neighborhood of $p$.  Then, there exists $M^\bullet \in D^b(\mathbb{Z})$ such that $\mathcal{F}^\bullet \cong M_Y^\bullet$ in $D^b(X;p)$.

Since $T_Y^*X \subseteq \pi^{-1}(Y)$, the previous proposition applies, and we may assume $\mathcal{F}^\bullet \cong f_* \mathcal{G}^\bullet$ in $D^b(X;p)$ for some $\mathcal{G}^\bullet \in D^b(Y)$.  In particular, this tells us $SS(\mathcal{F}^\bullet) = SS(f_*\mathcal{G}^\bullet)$ in a neighborhood of $p$.  So, the assumption that $SS(\mathcal{F}^\bullet) \subseteq T_Y^*X$, together with [5.4.4, K-S] (this tells us that $SS(f_*\mathcal{G}^\bullet)$ consists of covectors $(y,\xi) \in T^*X$ such that $y \in Y$, and $\xi \circ d_y f = \xi \in SS(\mathcal{G}^\bullet)$) give us that $SS(\mathcal{G}^\bullet)$ is contained in the zero section, $T_Y^*Y$ in a neighborhood of $p$:  Since $SS(\mathcal{G}^\bullet) \subseteq T^*Y$, the condition $SS(f_*\mathcal{G}^\bullet) \subseteq T_Y^*X$ implies $\xi = 0$) that $SS(\mathcal{G}^\bullet) \subseteq T_Y^*Y \cong Y \times \{0\}$ in a neighborhood of $\pi(p)$.

Next, let $g: Y \to \{pt\}$ be the canonical map to a point.  Since $SS(\mathcal{G}^\bullet) \subseteq Y \times \{0\}$, there exists a $M^\bullet \in D^b(\mathbb{Z})$ such that $\mathcal{G}^\bullet \cong g^* M^\bullet$.  Hence, $\mathcal{F}^\bullet \cong f_* g^* M^\bullet \cong M_Y^\bullet$ in $D^b(X;p)$, and we’re done.

Q.E.D.

Now, what can we say our constructible case $SS(\mathcal{F}^\bullet ) \subseteq \bigcup_S T_S^*X$?  Keywords: non-degenerate covectors, and Morse modules.  But that’s for another time.

## 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.