## 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.