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.