# The singular support of sheaves is $\gamma$-coisotropic

Stéphane Guillermou and Claude Viterbo

SG: UMR 5582 du CNRS Institut Fourier, Université Grenoble Alpes, CS 40700
38058 Grenoble cedex 9 - France
CV: Université Paris-Saclay, CNRS, Laboratoire de mathématiques d’Orsay,
91405, Orsay, France. Part of this paper was written as the author was a
member of DMA, École Normale Supérieure, 45 Rue d’Ulm, 75230 Cedex 05, FRANCE.
Both authors acknowledge support from ANR MICROLOCAL (ANR-15-CE40-0007) and
ANR COSY (ANR-21-CE40-0002).

,

###### Contents

  1. 1 Introduction
  2. 2 Notation, comments and acknowledgements
  3. 3 Reminders on the singular support of sheaves
  4. 4 Quantization of Hamiltonian isotopies
  5. 5 Spectral invariants for sheaves and Lagrangians
    1. 5.1 Definition of spectral invariants
    2. 5.2 The case of Hamiltonian maps
    3. 5.3 Defining $\gamma$-coisotropic subsets
  6. 6 The $\gamma_{g}$-metric on sheaves
    1. 6.1 The $\gamma_{g}$-topology
    2. 6.2 Link with spectral invariants
    3. 6.3 $\gamma_{g}$-limits and colimits
  7. 7 Proof of Theorem 1.2
  8. 8  $\gamma$-coisotropic vs cone-coisotropic
    1. 8.1 A $\gamma$-coisotropic set is cone-coisotropic
    2. 8.2 A cone-coisotropic set that is not $\gamma$-coisotropic
  9. 9 Questions and comments
  10. A  Persistence modules and Barcodes
  11. B Decomposition in the completion of $D_{lc}({\mathbb{R}})$
    1. B.1 Limits of constructible sheaves
    2. B.2 Decomposition in $D_{lc}(\mathbb{R})$
  12. C Comparing different interleaving metrics on $D(N\times{\mathbb{R}})$.
  13. D Local Floer cohomology

##  1\. Introduction

In [Vit92], a metric, denoted $\gamma$, was introduced on the set
${{\mathfrak{L}}}(T^{*}N)$ of Lagrangians Hamiltonianly isotopic to the zero
section in $T^{*}N$, where $N$ is a compact manifold and on
$\mathfrak{DHam}_{c}(T^{*}N)$ the group of Hamiltonian maps with compact
support for $N={\mathbb{R}}^{n}$ or $T^{n}$ (and in [Vit06] it was extended to
$\mathfrak{DHam}_{c}(T^{*}N)$ for general compact $N$). The metric was
generalized by Schwartz and Oh to general symplectic manifolds $(M,\omega)$
using Floer cohomology for the Hamiltonian case (see [Sch00, Oh05]), and by
[Lec08] for the Lagrangian case111Assuming $[\omega]\pi_{2}(M,L)=0$,
$\mu_{L}\pi_{2}(M,L)=0$, where $\mu_{L}$ is the Maslov class of $L$. . The
notion of $\gamma$-coisotropic sets in a symplectic manifold was defined in
[Vit22]. A similar definition was due to Usher [Ush19] in the setting of Hofer
distance under the name of “locally rigid”. Notice that this yields a weaker
notion : $\gamma$-coisotropic implies locally rigid in the sense of Usher.

In the context of sheaves on manifolds, there is a notion of coisotropic sets
in the sense of Kashiwara-Schapira (see [KS90], definition 6.5.1, p. 271 and
Definition 8.2), that the authors call involutivity. To avoid any confusion
with other notions, we shall use the term cone-coisotropic for their
definition as it is defined using the contingent and paratingent cones (see
[Bou32] ). With this notion, Kashiwara and Schapira proved

######  Theorem 1.1 ([KS90], theorem 6.5.4, p. 272).

Given a sheaf ${\mathcal{F}}\in D^{b}(N)$, its singular support
$SS({\mathcal{F}})$ is cone-coisotropic.

The aim of this paper is to prove that

######  Theorem 1.2.

Given a sheaf ${\mathcal{F}}\in D^{b}(N)$, its singular support
$SS({\mathcal{F}})$ is $\gamma$-coisotropic.

Theorem 1.2 implies Theorem 1.1, since we shall prove the following connection
between the two notions

######  Proposition 1.3.

Let $V$ be a closed subset in $(M,\omega)$. If $V$ is $\gamma$-coisotropic
then it is cone-coisotropic.

######  Remark 1.4.

This is related to questions asked by Vichery in section 4 of [Vic13]. Our
result, toghether with Proposition 1.3, stating that a $\gamma$-coisotropic
set is cone-coisotropic, gives a more natural proof of the Kashiwara-Schapira
theorem. Moreover, as opposed to the notion of cone-coisotropic, which is only
invariant by $C^{1}$ symplectic diffeomorphisms, the notion of
$\gamma$-coisotropic set is invariant by homeomorphisms preserving $\gamma$,
in particular by symplectic homeomorphisms, i.e. homeomorphisms which are
$C^{0}$ limits of symplectic diffeomorphisms. Along the way we prove a number
of results relating the singular support and the spectral norm $\gamma$.

##  2\. Notation, comments and acknowledgements

Since our results are local, we shall assume from now on that $N$ is a compact
manifold. The following definitions will turn out useful :

  * •

The category $D(N)$ is the unbounded derived category of sheaves of $k$-vector
spaces on $N$, for some given field $k$, but except in Appendix B the reader
may assume we are dealing with its bounded version. For the operations on
sheaves associated to a continuous map $f:M\longrightarrow N$ that is
$f^{-1},f_{*},f^{!},f_{!}$ and the two operations
$\otimes,\operatorname{\mathcal{H}om}$ and their derived versions,
$f^{-1},Rf_{*},Rf_{!},f^{!}$ and $\otimes,\operatorname{R\mathcal{H}om}$ on
$D(N)$ we refer to [KS90].

  * •

Choosing a real analytic structure on $N$ we set $D_{c}(N)$ to be the category
of constructible complexes ${\mathcal{F}}$, i.e. complexes such that, for some
subanalytic stratification, the restriction of ${\mathcal{F}}$ to each stratum
is locally constant and of finite rank.

  * •

For ${\mathcal{F}}\in D(N)$, $SS({\mathcal{F}})$ is the singular support (also
called microsupport) defined by Kashiwara-Schapira in [KS90]. We set
$SS^{\bullet}({\mathcal{F}})=SS({\mathcal{F}})\cap(T^{*}N\setminus 0_{N})$.

  * •

For an open set $U$ in the symplectic manifold $(M,\omega)$ we denote by
$\mathfrak{DHam}_{c}(U)$ the set of time one flows of Hamiltonian with compact
support contained in $U$.

We thank Pierre Schapira for useful conversations. While writing this paper,
we realized that some of the results in Subsection 6.3 were independently
discovered by Asano and Ike in [AI22].

##  3\. Reminders on the singular support of sheaves

Let $N$ be a manifold, $k$ a field and $D(N)$ the derived category of sheaves
of $k$-vector spaces on $N$. For ${\mathcal{F}}\in D(N)$ we recall that
$SS({\mathcal{F}})\subset T^{*}N$ is the closed conic subset defined by
Kashiwara-Schapira as the closure of the set of points $(x;\xi)$ such that
$(\operatorname{R\Gamma}_{\\{f\geq 0\\}}{\mathcal{F}})_{x}\not=0$ for some
function of class $C^{1}$ such that $f(x)=0$ and $df_{x}=\xi$. It is called
microsupport or singular support of ${\mathcal{F}}$.

The functor $\operatorname{R\Gamma}_{\\{f\geq 0\\}}$ does not commute with
infinite direct sums and $(-)_{x}$ does not commute with infinite direct
products. Here are two variations on the definition which have these
commutation properties. For the statement of the lemmas we introduce the
following definition.

######  Definition 3.1.

Let $\Omega\subset T^{*}N\setminus 0_{N}$ be an open conic subset. We call
$\Omega$-lens a locally closed subset $\Sigma$ of $N$ with the following
properties: $\overline{\Sigma}$ is compact and there exists an open
neighbourhood $U$ of $\overline{\Sigma}$ and a function $g\colon
U\times[0,1]\xrightarrow[]{}\mathbb{R}$ of class $C^{1}$ such that

  1. (1)

$dg_{t}(x)\in\Omega$ for all $(x,t)\in U\times[0,1]$, where
$g_{t}=g|_{U\times\\{t\\}}$,

  2. (2)

$\\{g_{t}<0\\}\subset\\{g_{t^{\prime}}<0\\}$ if $t\leq t^{\prime}$,

  3. (3)

the hypersurfaces $\\{g_{t}=0\\}$ coincide on $U\setminus\overline{\Sigma}$,

  4. (4)

$\Sigma=\\{g_{1}<0\\}\setminus\\{g_{0}<0\\}$.

We recall that $SS^{\bullet}(k_{\\{g_{t}<0\\}})=\\{(x;\lambda dg_{t}(x))$;
$g_{t}(x)=0$, $\lambda>0\\}$. Using the triangle
$k_{\\{g_{0}<0\\}}\xrightarrow[]{}k_{\\{g_{1}<0\\}}\xrightarrow[]{}k_{\Sigma}\xrightarrow[]{+1}$
and the “triangle inequality” for the singular support (see [KS90]) we obtain
$SS^{\bullet}(k_{\Sigma})\subset\Omega$.

\begin{overpic}[width=170.71652pt]{Lens-d.pdf} \put(78.0,35.0){$g_{1}=0$}
\put(17.0,45.0){$g_{0}=0$} \put(20.0,0.0){${\Omega}$} \end{overpic} Figure 1.
A lens. The arrows represent $\nabla g_{1}$ and $\nabla g_{0}$.

The next lemma is essentially a reformulation of the definition of the
singular support.

######  Lemma 3.2.

Let ${\mathcal{F}}\in D(N)$ and let $\Omega\subset T^{*}N\setminus 0_{N}$ be
an open conic subset. Then $SS({\mathcal{F}})\cap\Omega=\emptyset$ if and only
if $\operatorname{RHom}(k_{\Sigma},{\mathcal{F}})\simeq 0$ for any
$\Omega$-lens $\Sigma$.

###### Proof.

(i) We first assume $SS({\mathcal{F}})\cap\Omega=\emptyset$. We recall that
$\operatorname{RHom}(k_{V},{\mathcal{F}})\simeq\operatorname{R\Gamma}(V;{\mathcal{F}})$
for any open subset $V$ of $N$. Using the notations in the definition of
$\Omega$-lens we have a distinguished triangle
$k_{\\{g_{0}<0\\}}\xrightarrow[]{}k_{\\{g_{1}<0\\}}\xrightarrow[]{}k_{\Sigma}\xrightarrow[]{+1}$.
Applying $\operatorname{RHom}(-,{\mathcal{F}}|_{U})$ to this triangle we
obtain the result, using the non-characteristic deformation lemma ([KS90,
Prop. 2.7.2]) and the definition of the singular support.

(ii) We assume that there exists $(x_{0};\xi_{0})\in
SS({\mathcal{F}})\cap\Omega$. Hence we can find $(x_{1};\xi_{1})\in\Omega$
(close to $(x_{0};\xi_{0})$) and a function $f\colon
N\xrightarrow[]{}\mathbb{R}$ of class $C^{1}$ such that $f(x_{1})=0$,
$df(x_{1})=\xi_{1}$ and $(\operatorname{R\Gamma}_{\\{f\geq
0\\}}{\mathcal{F}})_{x_{1}}\not=0$. Now we can find a basis of open
neighbourhoods of $x_{1}$, say $U_{n}$, $n\in\mathbb{N}$, such that
$\Sigma_{n}:=\\{f\geq 0\\}\cap U_{n}$ is an $\Omega$-lens. Then there exists
$n$ such that $\operatorname{R\Gamma}(U_{n};\operatorname{R\Gamma}_{\\{f\geq
0\\}}{\mathcal{F}})\not=0$ and the result follows from

$\operatorname{R\Gamma}(U_{n};\operatorname{R\Gamma}_{\\{f\geq
0\\}}{\mathcal{F}})\simeq\operatorname{R\Gamma}(N;\operatorname{R\mathcal{H}om}(k_{U_{n}},\operatorname{R\mathcal{H}om}(k_{\\{f\geq
0\\}}{\mathcal{F}})))\\\
\simeq\operatorname{R\Gamma}(N;\operatorname{R\mathcal{H}om}(k_{\Sigma_{n}},{\mathcal{F}}))\simeq\operatorname{RHom}(k_{\Sigma_{n}},{\mathcal{F}})$

∎

The functor $\operatorname{RHom}(k_{\Sigma},-)$ commutes with direct products
but not with direct sums. Here is a dual version of the previous lemma using
the functor $\operatorname{R\Gamma}(N;k_{\Sigma}\otimes-)$ which commutes with
direct sums.

######  Lemma 3.3.

Let ${\mathcal{F}}\in D(N)$ and let $\Omega\subset T^{*}N\setminus 0_{N}$ be
an open conic subset. Then $SS({\mathcal{F}})\cap\Omega=\emptyset$ if and only
if $\operatorname{R\Gamma}(N;k_{\Sigma}\otimes{\mathcal{F}})\simeq 0$ for any
$\Omega^{a}$-lens $\Sigma$, where $\Omega^{a}$ is the antipodal set of
$\Omega$, that is, its image by $(x;\xi)\mapsto(x;-\xi)$.

The first author thanks Pierre Schapira for a suggestion simplifying the next
proof.

###### Proof.

Let $\Sigma$ be an $\Omega^{a}$-lens. We write it as the difference of the two
closed sets $\overline{\Sigma}$ and $Z=\overline{\Sigma}\setminus\Sigma$. We
can find two families of open neighbourhoods of $\overline{\Sigma}$, say
$\\{U_{n}\\}_{n\in\mathbb{N}}$, and $Z$, say $\\{V_{n}\\}_{n\in\mathbb{N}}$,
such that, for all $n$, we have: $V_{n}\subset U_{n}$ and $U_{n}\setminus
V_{n}$ is an $\Omega$-lens. For each $n$ we have the commutative diagram of
restriction maps

$\textstyle{\operatorname{R\Gamma}(U_{n};{\mathcal{F}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u_{n}}$$\textstyle{\operatorname{R\Gamma}(V_{n};{\mathcal{F}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{R\Gamma}(N;{\mathcal{F}}\otimes
k_{\overline{\Sigma}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\textstyle{\operatorname{R\Gamma}(N;{\mathcal{F}}\otimes
k_{Z})}$

The cone of $u$ is $\operatorname{R\Gamma}(N;k_{\Sigma}\otimes{\mathcal{F}})$,
so we want to prove that $u$ is an isomorphism, that is, $u$ induces an
isomorphism on all cohomology groups. The cone of $u_{n}$ is
$\operatorname{RHom}(k_{U_{n}\setminus V_{n}},F)$ and the previous lemma says
that $u_{n}$ is an isomorphism. Now the result follows from
$H^{i}(N;{\mathcal{F}}\otimes
k_{\overline{\Sigma}})\simeq\varinjlim_{n}H^{i}(U_{n};{\mathcal{F}})$ and
$H^{i}(N;{\mathcal{F}}\otimes k_{Z})\simeq\varinjlim_{n}H^{i}(V_{n};F)$, for
all $i$ (see [KS90, Rem. 2.6.6]). ∎

The previous two lemmas give the following proposition (see [KS90, Ex. 5.7]),
away from the zero-section. To deal with the zero-section we use
$SS({\mathcal{F}})\cap 0_{N}={\rm supp}({\mathcal{F}})$.

######  Proposition 3.4.

Let ${\mathcal{F}}_{n}\in D(N)$, $n\in\mathbb{N}$, be given. Then
$SS(\bigoplus_{n}{\mathcal{F}}_{n})\subset\overline{\bigcup_{n}SS({\mathcal{F}}_{n})}$
and
$SS(\prod_{n}{\mathcal{F}}_{n})\subset\overline{\bigcup_{n}SS({\mathcal{F}}_{n})}$.

##  4\. Quantization of Hamiltonian isotopies

Everything in the next subsection is from [GKS] or [KS90]. Given $\varphi$, a
homogeneous Hamiltonian map from $T^{*}N\setminus 0_{N}$ to $T^{*}N\setminus
0_{N}$, Guillermou, Kashiwara and Schapira associate a Kernel, that is an
element of $D^{b}(N\times N)$, noted $K_{\varphi}$ having the following
properties222In fact in [GKS] the authors associate to a Hamiltonian isotopy
$\varphi^{s}$ a Kernel ${\mathcal{K}}_{\Phi}\in D^{b}(N\times N\times[0,1])$
such that ${{\mathcal{K}}_{\Phi}}_{\mid N\times
N\times\\{s\\}}=K_{\varphi^{s}}$.. First $K_{\varphi}$ is a quantization of
the homogeneous Lagrangian $\Lambda_{\varphi}=\\{(z,\varphi(z))\mid z\in
T^{*}N\\}\subset{\overline{T^{*}N}}\times T^{*}N$ in the sense that
$SS^{\bullet}(K_{\varphi})=\Lambda_{\varphi}$, where we set for short, for a
sheaf ${\mathcal{F}}$ on a manifold $N$,

$SS^{\bullet}({\mathcal{F}})=SS({\mathcal{F}})\cap(T^{*}N\setminus 0_{N})$

Before we state the second property we recall the composition of sheaves:

######  Definition 4.1.

Let ${\mathcal{F}}\in D^{b}(M)$ and $K\in D^{b}(M\times N)$. We define

$K({\mathcal{F}})={\mathcal{F}}\circ K=R{q_{N}}_{!}(K\otimes
q_{M}^{-1}{\mathcal{F}})\in D^{b}(N),$

where $q_{M},q_{N}$ are the projections of $M\times N$ on $M$ and $N$
respectively.

The second property is then :

######  Proposition 4.2 (See Proposition 7.1.2 (ii) in [KS90] and [GKS],
formula (1.12) ).

We have

$SS^{\bullet}(K_{\varphi}({\mathcal{F}}))=\varphi(SS^{\bullet}({\mathcal{F}}))$

Now if $\varphi$ is an exact non-homogeneous symplectic map from $T^{*}M$ to
$T^{*}N$ (i.e. $PdQ-pdq$ is exact), we replace $\Lambda_{\varphi}$ by the
homogeneous lift of the graph of $\varphi$, say
$\widehat{\Lambda}_{\varphi}\subset\overline{T^{*}M}\times T^{*}N\times
T^{*}{\mathbb{R}}$, given by

$\widehat{\Lambda}_{\varphi}=\left\\{(q,-\tau p,Q(q,p),\tau
P(q,p),F(q,p),\tau)\mid\varphi(q,p)=(Q,P),\tau>0\right\\}$

where $dF=PdQ-pdq$. Note that considering $\widehat{\Lambda}_{\varphi}$ as a
correspondence in $\overline{T^{*}M}\times T^{*}N\times T^{*}{\mathbb{R}}$, if
$L$ is an exact Lagrangian in $T^{*}M$ and

$\widehat{L}=\left\\{(q,\tau p,f_{L}(q,p),\tau)\mid(q,p)\in L,\tau\geq
0\right\\}$

we have $\widehat{\Lambda}_{\varphi}\circ\widehat{L}=\widehat{\varphi(L)}$.

From now on we assume $M=N$ and $\varphi$ is a Hamiltonian map with compact
support (which will always be the case in this paper), the existence of a
quantization ${\mathcal{K}}_{\varphi}$ for $\widehat{\Lambda}_{\varphi}$
follows from the homogeneous case by considering $\widehat{\Lambda}_{\varphi}$
as a deformation of $\widehat{\Lambda}_{\rm id}$. Since
$\widehat{\Lambda}_{\rm id}$ is quantized by $k_{\Delta_{N}\times[0,+\infty[}$
we conclude by Proposition 4.2. However we have to be careful that the
singular support is slightly bigger than desired:

$SS^{\bullet}(k_{\Delta_{N}\times[0,+\infty[})=\widehat{\Lambda}_{\rm
id}\cup\\{(q,p,q,-p,t,0)\mid t\geq 0\\}$

but its intersection with $\\{\tau>0\\}$ is equal to $\widehat{\Lambda}_{{\rm
id}}$. We can find a homogeneous map $\psi$ acting on
$T^{*}(N^{2}\times\mathbb{R})\setminus 0_{N^{2}\times\mathbb{R}}$ such that
$\psi$ is the identity map near $\\{\tau=0\\}$ and
$\psi(\widehat{\Lambda}_{\rm id})=\widehat{\Lambda}_{\varphi}$. We remark that
$\varphi$ can be lifted to a homogeneous map $\widehat{\varphi}$ acting on
$T^{*}(N\times\mathbb{R})\setminus\\{\tau=0\\}$ (we homogenize the Hamiltonian
function) but we cannot use ${\rm id}_{T^{*}N}\times\widehat{\varphi}$ for
$\psi$, as would be natural, since it is not defined everywhere outside the
zero section.

Now we set

${\mathcal{K}}_{\varphi}=K_{\psi}(k_{\Delta_{N}\times[0,+\infty[})$

and we find

$SS^{\bullet}({\mathcal{K}}_{\varphi})=\widehat{\Lambda}_{\varphi}\cup\\{(q,p,q,-p,t,0)\mid
t\geq 0\\}$

If we consider the whole isotopy $\varphi_{s}$ and choose $t_{0}$ big enough
so that $\widehat{\Lambda}_{\varphi_{s}}$ is contained in
$T^{*}(N^{2}\times\mathopen{]}-t_{0},t_{0}[)$ for all $s\in[0,1]$, then we see
that $SS^{\bullet}({\mathcal{K}}_{\varphi_{s}})\cap T^{*}U$ is independent of
$s$, where $U=N^{2}\times(\mathbb{R}\setminus[-t_{0},t_{0}])$. By [KS90, Prop.
5.4.5] we deduce that

${\mathcal{K}}_{\varphi}|_{N^{2}\times]-\infty,-t_{0}[}\simeq
0,\qquad{\mathcal{K}}_{\varphi}|_{N^{2}\times]t_{0},+\infty[}\simeq
k_{\Delta_{N}\times[0,+\infty[}|_{N^{2}\times]t_{0},+\infty[}$

In other words ${\mathcal{K}}_{\varphi}={\mathcal{K}}_{\rm
id}=k_{\Delta_{N}\times[0,+\infty[}$ outside of a compact set.

######  Remark 4.3.

We could obtain the existence of ${\mathcal{K}}_{\varphi}$ when $M=N$ and
$\varphi$ has compact support without assuming that $\varphi$ is Hamiltonian
by adapting the result of [Gui12, Vit19] (see Theorem 5.1). In loc. cit. the
existence of a quantization is proved for a Legendrian of $J^{1}N$ which is
the lift of a compact exact Lagrangian of $T^{*}N$. Here the graph of
$\varphi$ is not compact but coincides with the diagonal
$T^{*}_{\Delta_{N}}N^{2}$ outside a compact set (if we choose $F=0$ at
infinity) so it should not be too difficult to extend the result in our
situation.

Since we use ${\mathcal{K}}_{\varphi}$, defined over $N^{2}\times\mathbb{R}$,
instead of sheaf defined over $(N\times\mathbb{R})^{2}$, we change slightly
the composition functor as follows, which becomes a mixture of $\circ$ and the
convolution $\ast$ (see Section 5 for the definition)

######  Definition 4.4.

Let ${\mathcal{F}}\in D^{b}(M\times\mathbb{R})$ and ${\mathcal{K}}\in
D^{b}(M\times N\times\mathbb{R})$. We define

${\mathcal{K}}^{\varoast}({\mathcal{F}})={\mathcal{F}}\varoast{\mathcal{K}}=Rs_{!}R{q_{2}}_{!}(q_{1,2}^{-1}{\mathcal{K}}\otimes
q_{1}^{-1}{\mathcal{F}})\in D^{b}(N\times{\mathbb{R}}),$

where the maps $q_{1,2}\colon M\times
N\times{\mathbb{R}}^{2}\xrightarrow[]{}M\times N\times{\mathbb{R}}$,
$q_{1}\colon M\times
N\times{\mathbb{R}}^{2}\xrightarrow[]{}M\times{\mathbb{R}}$, $s\colon M\times
N\times{\mathbb{R}}^{2}\xrightarrow[]{}N\times{\mathbb{R}}$ are defined by
$q_{1,2}(x,y,t_{1},t_{2})=(x,y,t_{1})$, $q_{1}(x,y,t_{1},t_{2})=(x,t_{2})$,
$s(x,y,t_{1},t_{2})=(x,y,t_{1}+t_{2})$.

Now we obtain a non homogeneous version of Proposition 4.2.

######  Proposition 4.5.

For a compactly supported Hamiltonian map $\varphi$ from $T^{*}N$ to itself
and any ${\mathcal{F}}\in D^{b}(N\times\mathbb{R})$ with
$SS({\mathcal{F}})\subset\\{\tau\geq 0\\}$, we have
$SS^{\bullet}({\mathcal{K}}_{\varphi}^{\varoast}({\mathcal{F}}))=\widehat{\varphi}(SS^{\bullet}({\mathcal{F}}))$.

In particular, for $L\in\mathcal{L}(T^{*}N)$ we have
$SS^{\bullet}({\mathcal{K}}_{\varphi}({\mathcal{F}}_{L}))=SS^{\bullet}({\mathcal{F}}_{\varphi(L)})$
and, by uniqueness of the quantization (see Theorem 5.1, (4)),
${\mathcal{K}}_{\varphi}^{\varoast}({\mathcal{F}}_{L})\simeq{\mathcal{F}}_{\varphi(L)}$.

##  5\. Spectral invariants for sheaves and Lagrangians

For $M$ a symplectic manifold, (here we only need $M=T^{*}N$), if $\Lambda(M)$
is the bundle of Lagrangians subspaces of the tangent bundle to $M$, with
fibre the Lagrangian Grassmannian $\Lambda(T_{z}M)\simeq\Lambda(n)$, we denote
by $\widetilde{\Lambda}(M)$ the bundle induced by the universal cover
$\widetilde{\Lambda}(n)\longrightarrow\Lambda(n)$.

When using coefficients others than $\mathbb{Z}/2\mathbb{Z}$ we assume we have
a lifting $\widetilde{G}_{L}$ of the Gauss map
$G_{L}:L\longrightarrow\Lambda(T^{*}N)$ given by $x\mapsto T_{x}L$ to
$\widetilde{\Lambda}(T^{*}N)$. This is called a grading. Given a graded $L$,
the canonical automorphism of the covering induces a new grading and we denote
it as $L[1]$ (or $\widetilde{L}[1]$), and its $k$-th iteration as $L[k]$ (or
$\widetilde{L}[k]$). The grading yields an absolute grading for the Floer
homology333Without this extra piece of information, the Floer cohomology
$FH^{k}(L_{1},L_{2})$ is generally defined only up to a shift in grading.
However, when $L_{2}=\varphi_{H}^{1}(L_{1})$, there is an absolute grading, a
priori depending on the choice of $H$. However in our situation we do not
assume $L_{1}$ is Hamiltonianly isotopic to $L_{2}$, so the grading is
required to have an absolute grading of the Floer cohomology. and hence for
the complex of sheaves in the Theorem stated below. We refer to [Sei00] for
more details on this, but point out that we shall never mention explicitly the
grading.

An exact Lagrangian in $T^{*}N$ is a pair $(L,f_{L})$ such that
$df_{L}=\lambda_{\mid L}$ where $\lambda=pdq$ is the Liouville form. For an
exact Lagrangian, a grading always exists since the obstruction to its
existence is given by the Maslov class, and for exact Lagrangians in $T^{*}N$
the Maslov class vanishes, as was proved by Kragh and Abouzaid (see [Kra13]
and also the sheaf-theoretic proof by [Gui12]). When $f_{L}$ is implicit we
only write $L$, for example $0_{N}$ means $(0_{N},0)$. A Lagrangian brane is a
triple $\widetilde{L}=(L,f_{L},\widetilde{G}_{L})$, where $L$ is a compact
Lagrangian. For $c$ a real constant, we write
$T_{c}\widetilde{L}=(L,f_{L}+c,\widetilde{G}_{L})$. We also use the notation
$\widetilde{L}+c$ or $L+c$ if the grading is irrelevant (as it will be most of
the time).

Let $L$ be a Lagrangian in $T^{*}N$ and

$\widehat{L}=\left\\{(q,\tau p,f_{L}(q,p),\tau)\mid(q,p)\in L,\tau\geq
0\right\\}$

the homogenized Lagrangian in $T^{*}(N\times{\mathbb{R}})$.

Similarly for Hamiltonian maps in an exact manifold $(M,\omega=d\lambda)$ we
look for pairs $(\varphi,F)$ such that $\varphi^{*}\lambda-\lambda=dF$ on $M$.

To state the next result we recall the two adjoint functors, $\ast$ and
$\operatorname{\mathcal{H}om}^{\ast}$, on the category $D(N\times\mathbb{R})$
introduced by Tamarkin in [Tam08]. First we set
$s,q_{1},q_{2}:N\times({\mathbb{R}})^{2}\longrightarrow N\times{\mathbb{R}}$
given by $s(x,t_{1},t_{2})=(x,t_{1}+t_{2})$, $q_{1}(x,t_{1},t_{2})=(x,t_{1})$,
$q_{2}(x,t_{1},t_{2})=(x,t_{2})$ and we define

$\displaystyle{\mathcal{F}}\ast{\mathcal{G}}$
$\displaystyle=Rs_{!}(q_{1}^{-1}{\mathcal{F}}\otimes
q_{2}^{-1}{\mathcal{G}}),$
$\displaystyle\operatorname{\mathcal{H}om}^{\ast}({\mathcal{F}},{\mathcal{G}})$
$\displaystyle=Rq_{1*}\operatorname{R\mathcal{H}om}(q_{2}^{-1}{\mathcal{F}},s^{!}{\mathcal{G}})$

Then $\operatorname{\mathcal{H}om}^{\ast}$ is the adjoint of $\ast$ in the
sense that

${\rm
Mor}_{D(N\times{\mathbb{R}})}({\mathcal{F}},\operatorname{\mathcal{H}om}^{\ast}({\mathcal{G}},\mathcal{H}))={\rm
Mor}_{D(N\times{\mathbb{R}})}({\mathcal{F}}\ast{\mathcal{G}},\mathcal{H})$

We recall some properties of $\operatorname{\mathcal{H}om}^{\ast}$. We denote
by $D_{\tau\geq 0}(N\times\mathbb{R})$ the subcategory of
$D(N\times\mathbb{R})$ of objects with singular support contained444This
category contains the “Tamarkin category” which is the left orthogonal of
$D_{\tau\leq 0}(N\times\mathbb{R})$. The sheaves ${\mathcal{F}}_{L}$ belongs
to the Tamarkin category. The category $D_{\tau\geq 0}(N\times\mathbb{R})$ is
stable by $\ast$ and $\operatorname{\mathcal{H}om}^{\ast}$. The Tamarkin
category is stable by $\ast$ but not $\operatorname{\mathcal{H}om}^{\ast}$:
typically when $N$ is a point we have
$\operatorname{\mathcal{H}om}^{\ast}(k_{[a,\infty[},k_{[b,\infty[})\simeq
k_{]-\infty,b-a[}[1]$. in $\\{\tau\geq 0\\}$. A base change formula gives, for
any $c\in\mathbb{R}$, and ${\mathcal{F}},{\mathcal{G}}\in
D^{b}(N\times{\mathbb{R}})$

(5.1)
$\operatorname{R\Gamma}_{N\times\\{-c\\}}(N\times\mathbb{R};\operatorname{\mathcal{H}om}^{\ast}({\mathcal{F}},{\mathcal{G}}))\simeq\operatorname{RHom}({\mathcal{F}},T_{c*}({\mathcal{G}})),$

with $T_{c}\colon N\times\mathbb{R}\xrightarrow[]{}N\times\mathbb{R}$,
$(x,t)\mapsto(x,t+c)$. Note that $\widehat{T_{c}L}=T_{c}\widehat{L}$.

For $\mathcal{H}$ with singular support in $\\{\tau\geq 0\\}$, we have a
morphism

$\operatorname{R\Gamma}_{N\times\\{b\\}}(N\times\mathbb{R};\mathcal{H})\xrightarrow[]{}\operatorname{R\Gamma}_{N\times[a,b]}(N\times\mathbb{R};\mathcal{H})\xleftarrow[]{{\raisebox{-2.58334pt}[0.0pt][-2.58334pt]{$\mspace{1.0mu}\sim\mspace{2.0mu}$}}}\operatorname{R\Gamma}_{N\times\\{a\\}}(N\times\mathbb{R};\mathcal{H})$

for any $a\leq b$. We remark that the cone of this morphism is
$\operatorname{R\Gamma}_{N\times[a,b[}(N\times\mathbb{R};\mathcal{H})$. Using
(5.1) we deduce a morphism, say $u$, from
$\operatorname{RHom}({\mathcal{F}},T_{c*}({\mathcal{G}}))$ to
$\operatorname{RHom}({\mathcal{F}},T_{d*}({\mathcal{G}}))$ for any $c\leq d$.
Taking ${\mathcal{F}}={\mathcal{G}}$, $c=0$ the image of ${\rm
id}_{\mathcal{G}}$ gives

(5.2)
$\tau_{0,d}\colon{\mathcal{G}}\xrightarrow[]{}T_{d*}({\mathcal{G}})\qquad\text{for
any $d\geq 0$}$

(we come back to this in §6). Then $u$ is nothing but the composition with
$T_{c*}(\tau_{0,d-c})$.

According to [Gui12] and [Vit19] we have

######  Theorem 5.1.

To each $L\in\mathcal{L}(T^{*}N)$ we can associate ${\mathcal{F}}_{L}\in
D^{b}(N\times{\mathbb{R}})$ such that

  1. (1)

$SS({\mathcal{F}}_{L})=\widehat{L}$.

  2. (2)

${\mathcal{F}}_{L}$ is simple (cf. [KS90, Def. 7.5.4]), ${\mathcal{F}}_{L}=0$
near $N\times\\{-\infty\\}$ and ${\mathcal{F}}_{L}=k_{N\times\mathbb{R}}$ near
$N\times\\{+\infty\\}$.

  3. (3)

We have an isomorphism

$FH^{\bullet}(L_{0},L_{1};a,b)=H^{*}_{N\times[a,b[}\left(N\times\mathbb{R};{\mathcal{H}om}^{\ast}({\mathcal{F}}_{L_{0}},{\mathcal{F}}_{L_{1}})\right)$

  4. (4)

${\mathcal{F}}_{L}$ is unique satisfying properties (1) and (2).

  5. (5)

There is a natural product map

$\operatorname{\mathcal{H}om}^{\ast}({\mathcal{F}}_{L_{1}},{\mathcal{F}}_{L_{2}})\otimes\operatorname{\mathcal{H}om}^{\ast}({\mathcal{F}}_{L_{2}},{\mathcal{F}}_{L_{3}})\longrightarrow\operatorname{\mathcal{H}om}^{\ast}({\mathcal{F}}_{L_{1}},{\mathcal{F}}_{L_{3}})$

inducing in cohomology a map

$\displaystyle
H^{*}_{N\times[\lambda,+\infty[}(N\times\mathbb{R};\operatorname{\mathcal{H}om}^{\ast}({\mathcal{F}}_{L_{1}},{\mathcal{F}}_{L_{2}}))\otimes
H^{*}_{N\times[\mu,+\infty[}(N\times\mathbb{R};\operatorname{\mathcal{H}om}^{\ast}({\mathcal{F}}_{L_{2}},{\mathcal{F}}_{L_{3}}))$
$\displaystyle\Big{\downarrow}\cup_{\ast}$ $\displaystyle
H^{*}_{N\times[\lambda+\mu,+\infty[}(N\times\mathbb{R};\operatorname{\mathcal{H}om}^{\ast}({\mathcal{F}}_{L_{1}},{\mathcal{F}}_{L_{3}}))$

that coincides through the above identifications with the triangle product in
Floer cohomology.

######  Remarks 5.2.

  1. (1)

Note that we have

${\mathcal{F}}_{T_{c}L}={T_{c}}_{*}({\mathcal{F}}_{L})$

  2. (2)

In case $L$ has a Generating Function Quadratic at infinity, $S(x,\xi)$
defined on the vector bundle $E\overset{\pi}{\longrightarrow}N$, we can take
for ${\mathcal{F}}_{L}$ the complex $R(\pi\times{\rm
id}_{\mathbb{R}})_{*}(k_{U_{S}})$ where $U_{S}=\\{(x,\xi,t)\mid S(x,\xi)\leq
t\\}$.

###  5.1. Definition of spectral invariants

Now we can give a sheafy definition of the spectral invariants introduced in
[Vit92]. The next formulation appears in [Vic13] (see also [Vit22]).

In fact the invariants can be defined for objects ${\mathcal{F}}$ of
$D_{\tau\geq 0}(N\times\mathbb{R})$ satisfying:

(5.3) ${\mathcal{F}}|_{N\times]-\infty,-t_{0}[}\simeq
0,\quad{\mathcal{F}}|_{N\times]t_{0},+\infty[}\simeq
k_{N\times\mathbb{R}}|_{N\times]t_{0},+\infty[}\quad\text{for some $t_{0}$.}$

We have seen that ${\mathcal{F}}_{L}$ satisfies (5.3) for any
$L\in\mathcal{L}(T^{*}N)$. For ${\mathcal{F}}_{1}$, ${\mathcal{F}}_{2}\in
D_{\tau\geq 0}(N\times\mathbb{R})$ satisfying (5.3),
$\operatorname{\mathcal{H}om}^{\ast}({\mathcal{F}}_{1},{\mathcal{F}}_{2}))$ is
$k_{N\times\mathbb{R}}[1]$ near $N\times\\{-\infty\\}$ and $0$ near
$N\times\\{+\infty\\}$. Using 5.1 and
$H^{0}_{N\times\\{c\\}}(N\times\mathbb{R};k_{N\times\mathbb{R}}[1])\simeq k$,
we deduce a natural morphism
${\mathcal{F}}_{1}\xrightarrow[]{}T_{c*}({\mathcal{F}}_{2})$, for any $c$ big
enough, which restricts to ${\rm id}\colon
k_{N\times\mathbb{R}}\xrightarrow[]{}k_{N\times\mathbb{R}}$ near
$N\times\\{+\infty\\}$. More generally
$H^{*}_{N\times\\{c\\}}(N\times\mathbb{R};k_{N\times\mathbb{R}}[1])\simeq
H^{*}(N;k_{N})$ and any class $\alpha\in H^{d}(N;k_{N})$ yields a morphism

$u(\alpha,{\mathcal{F}}_{1},{\mathcal{F}}_{2},c)\colon{\mathcal{F}}_{1}\xrightarrow[]{}T_{c*}({\mathcal{F}}_{2})[d],$

for $c$ large enough, which restricts to $\alpha\colon
k_{N\times\mathbb{R}}\xrightarrow[]{}k_{N\times\mathbb{R}}[d]$ near
$N\times\\{+\infty\\}$ (using ${\rm Mor}(k_{N},k_{N}[d])\simeq
H^{d}(N;k_{N})$). The cup product corresponds to the composition, and for $c$
(and $t_{0}$) large enough this is summarized by the following diagram

$\textstyle{H^{*}(N,k)\simeq
R\Gamma(N,k_{N})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\textstyle{\operatorname{RHom}({\mathcal{F}}_{1},{T_{c}}_{*}{\mathcal{F}}_{2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\operatorname{RHom}(k_{N\times]t_{0},+\infty[},k_{N\times]t_{0},+\infty[})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\operatorname{RHom}\left({{\mathcal{F}}_{1}}_{\mid
N\times]t_{0},+\infty[},{T_{c}}_{*}{{\mathcal{F}}_{2}}_{\mid
N\times]t_{0},+\infty[}\right)}$

######  Definition 5.3.

For ${\mathcal{F}}_{1}$, ${\mathcal{F}}_{2}\in D_{\tau\geq
0}(N\times\mathbb{R})$ satisfying (5.3) and $\alpha\in H^{d}(N;k_{N})$ we set

$c(\alpha,{\mathcal{F}}_{1},{\mathcal{F}}_{2})=-\inf\\{c;\;\exists
u\colon{\mathcal{F}}_{1}\xrightarrow[]{}T_{c*}({\mathcal{F}}_{2})[d],\;u|_{N\times]t_{0},+\infty[}=\alpha,\;\text{for
$t_{0}\gg 0$}\\}$

(Here $t_{0}$ is big enough so that both ${\mathcal{F}}_{1}$ and
$T_{c*}({\mathcal{F}}_{2})$ satisfy (5.3)). We also set
$c_{-}({\mathcal{F}}_{1},{\mathcal{F}}_{2})=c(1,{\mathcal{F}}_{1},{\mathcal{F}}_{2})$,
if $N$ is oriented with fundamental class $\mu_{N}$,
$c_{+}({\mathcal{F}}_{1},{\mathcal{F}}_{2})=c(\mu_{N},{\mathcal{F}}_{1},{\mathcal{F}}_{2})$
and
$\gamma({\mathcal{F}}_{1},{\mathcal{F}}_{2})=c_{+}({\mathcal{F}}_{1},{\mathcal{F}}_{2})-c_{-}({\mathcal{F}}_{1},{\mathcal{F}}_{2})$.

For $L_{1},L_{2}\in\mathcal{L}(T^{*}N)$ we set
$c(\alpha,L_{1},L_{2})=c(\alpha,{\mathcal{F}}_{L_{1}},{\mathcal{F}}_{L_{2}})$
and define $c_{\pm}(L_{1},L_{2})$, $\gamma(L_{1},L_{2})$ in the same way.
Finally we set $c_{\pm}(L)=c_{\pm}(0_{N},L)$.

It is not difficult to see that

$c_{-}({\mathcal{F}}_{1},{\mathcal{F}}_{2})=\inf\\{c(\alpha,{\mathcal{F}}_{1},{\mathcal{F}}_{2});\;\alpha\in
H^{*}(N;k_{N})\\}$

and

$c_{+}({\mathcal{F}}_{1},{\mathcal{F}}_{2})=\sup\\{c(\alpha,{\mathcal{F}}_{1},{\mathcal{F}}_{2});\;\alpha\in
H^{*}(N;k_{N})\\}$

Indeed, a class $\alpha$ induces by multiplication
$v\colon{\mathcal{F}}_{2}\xrightarrow[]{}{\mathcal{F}}_{2}[d],\;v|_{N\times]t_{0},+\infty[}=\alpha$;
now if we have
$u\colon{\mathcal{F}}_{1}\xrightarrow[]{}T_{c*}({\mathcal{F}}_{2})[0],\;u|_{N\times]t_{0},+\infty[}=1$,
then $v\circ
u\colon{\mathcal{F}}_{1}\xrightarrow[]{}T_{c*}({\mathcal{F}}_{2})[d]$
satisfies $v\circ u|_{N\times]t_{0},+\infty[}=\alpha$ so
$c(1,{\mathcal{F}}_{1},{\mathcal{F}}_{2})\leq
c(\alpha,{\mathcal{F}}_{1},{\mathcal{F}}_{2})$.

Using the construction of the sheaf ${\mathcal{F}}_{L}$ by using Floer
cohomology it is easy to see that the present definition of
$c(\alpha,{\mathcal{F}}_{L_{1}},{\mathcal{F}}_{L_{2}})$ coincides with the
definition of $c(\alpha,L_{1},L_{2})$ using Floer cohomology. In particular,
we have that
$c_{+}({\mathcal{F}}_{L_{1}},{\mathcal{F}}_{L_{2}})=-c_{-}({\mathcal{F}}_{L_{2}},{\mathcal{F}}_{L_{1}})$
and this can be extended to the general case of
${\mathcal{F}}_{1},{\mathcal{F}}_{2}$ using [Vit19].

We have given the definition using the right hand side of (5.1). We can also
rewrite it with the left hand side as follows. Let us set
$\mathcal{H}=Rt_{*}\operatorname{\mathcal{H}om}^{\ast}({\mathcal{F}},{\mathcal{G}}))$,
where $t\colon N\times\mathbb{R}\xrightarrow[]{}\mathbb{R}$ is the projection.
Then $\mathcal{H}$ contains all the information about the spectral invariants.
More precisely (5.1) is isomorphic to
$\operatorname{R\Gamma}_{\\{-c\\}}(\mathcal{H})$ and $\mathcal{H}$ has its
singular support in $\\{\tau\geq 0\\}$ so we have the morphisms

(5.4)
$\operatorname{R\Gamma}(N;k_{N})\simeq\operatorname{R\Gamma}_{\\{-t_{0}\\}}(\mathbb{R};\mathcal{H})\xrightarrow[]{{\raisebox{-2.58334pt}[0.0pt][-2.58334pt]{$\mspace{1.0mu}\sim\mspace{2.0mu}$}}}\operatorname{R\Gamma}_{[-t_{0},-c]}(\mathbb{R};\mathcal{H})\xleftarrow[]{}\operatorname{R\Gamma}_{\\{-c\\}}(\mathbb{R};\mathcal{H})$

for $t_{0}\gg 0$ and any $c\leq t_{0}$. To find the spectral invariants we
look for the infimum on the $c$ such that a given class is in the image of the
composition of this morphisms.

######  Remark 5.4.

We can see that the infimum in Definition 5.3 is actually a minimum, at least
when ${\mathcal{F}}$ is of the form ${\mathcal{F}}_{L}$. Indeed, this follows
either by identifying the spectral invariants of the Definition to those
obtained by using Floer cohomology, or by appealing to the theory of
persistence modules and barcodes (see Appendix A Proposition A.1).

We notice the following immediate consequence of the definitions

######  Proposition 5.5.

There exists a morphism
$u:{\mathcal{F}}_{L_{1}}\longrightarrow{\mathcal{F}}_{L_{2}}$ such that
$u={\rm Id}$ at $+\infty$ if and only if $c_{-}(L_{1},L_{2})\geq 0$. In
particular there is a morphism
$k_{N\times[0,+\infty[}\longrightarrow{\mathcal{F}}$ if and only if
$c_{-}(L)\geq 0$.

###  5.2. The case of Hamiltonian maps

If $\varphi$ is a Hamiltonian map of $T^{*}\mathbb{R}^{n}$ with compact
support, we define the spectral invariants of $\varphi$ as those of its graph,
say $\Gamma_{\varphi}\subset\overline{T^{*}\mathbb{R}^{n}}\times
T^{*}\mathbb{R}^{n}$, compactified in the following way: we choose a
symplectomorphism $\psi\colon\overline{T^{*}\mathbb{R}^{n}}\times
T^{*}\mathbb{R}^{n}\xrightarrow[]{{\raisebox{-2.58334pt}[0.0pt][-2.58334pt]{$\mspace{1.0mu}\sim\mspace{2.0mu}$}}}T^{*}\Gamma_{{\rm
id}}$ such that $\psi(\Gamma_{{\rm id}})=0_{\Gamma_{{\rm id}}}$; then
$\psi(\Gamma_{\varphi})$ coincides with the zero-section outside a compact set
and we can compactify it as $\Gamma_{\varphi}^{c}$ in the cotangent bundle of
the sphere. In particular $\Gamma_{\rm id}^{c}=0_{S^{2n}}$ and the invariants
of $\varphi$, $c_{+}(\varphi)$, $c_{-}(\varphi)$, are defined as those of
$(\Gamma_{\varphi}^{c},0_{S^{2n}})$. This is well-defined since another choice
of $\psi$ would give a pair $(\Gamma_{\varphi}^{\prime c},0_{S^{2n}})$
symplectomorphic to $(\Gamma_{\varphi}^{c},0_{S^{2n}})$.

On the other hand we have seen in §4 that $\varphi$ has an associated sheaf
${\mathcal{K}}_{\varphi}\in D(\mathbb{R}^{2n+1})$. In the next paragraph we
explain how we can turn the above definition of $c_{+}(\varphi)$,
$c_{-}(\varphi)$ into Definition 5.6 which uses ${\mathcal{K}}_{\varphi}$,
${\mathcal{K}}_{\rm id}$ and not the intermediate symplectomorphism $\psi$.

#### Equivalence with Definition 5.6

We first remark that we don’t really need to compactify $\Gamma_{\rm id}$ into
the sphere to obtain the invariants. Let $D_{1}$ be the subcategory of
$D_{\tau\geq 0}(\Gamma_{\rm id}\times\mathbb{R})$ of complexes isomorphic to
$k_{\Gamma_{\rm id}\times[c,+\infty[}$ (for some $c$) and $D_{2}$ be the
subcategory of $D_{\tau\geq 0}(S^{2n}\times\mathbb{R})$ of complexes
isomorphic to $k_{S^{2n}\times[c,+\infty[}$ near
$\\{\infty\\}\times[c,+\infty[$. Then the compactification and the restriction
map induce functors $\alpha\colon D_{1}\xrightarrow[]{}D_{2}$ and $\beta\colon
D_{2}\xrightarrow[]{}D_{1}$ such that $\beta\circ\alpha={\rm id}$,
$\alpha(\tau_{a,b}({\mathcal{F}}))=\tau_{a,b}(\alpha({\mathcal{F}}))$. We
deduce that
$c_{-}(\alpha({\mathcal{F}}_{1}),\alpha({\mathcal{F}}_{2}))=c_{-}({\mathcal{F}}_{1},{\mathcal{F}}_{2})$
(we remark that $c_{-}$ makes sense even in the non compact case). In
particular

(5.5)
$c_{-}(\varphi)=c_{-}({\mathcal{F}}_{\Gamma_{\varphi}^{c}},{\mathcal{F}}_{0_{S^{2n}}})=c_{-}({\mathcal{F}}_{\Gamma_{\varphi}},{\mathcal{F}}_{0_{\mathbb{R}^{2n}}})$

Now the symplectomorphism $\psi$ induces an equivalence of categories between
${\mathcal{K}}_{\psi}^{\varoast}\colon D_{\tau\geq
0}(\mathbb{R}^{2n+1})\xrightarrow[]{{\raisebox{-2.58334pt}[0.0pt][-2.58334pt]{$\mspace{1.0mu}\sim\mspace{2.0mu}$}}}D_{\tau\geq
0}(\Gamma_{\rm id}\times\mathbb{R})$ which commutes with the translation
$T_{c*}$ in the last variables. We have
${\mathcal{K}}_{\psi}^{\varoast}({\mathcal{K}}_{\varphi}))\simeq{\mathcal{F}}_{\Gamma_{\varphi}}$
and we obtain from (5.5) that the spectral invariant $c_{-}(\varphi)$ we have
defined above coincides with the one in Definition 5.6 below. To obtain
$c_{+}$ we use
$c_{+}({\mathcal{F}}_{1},{\mathcal{F}}_{2})=-c_{-}({\mathcal{F}}_{2},{\mathcal{F}}_{1})$.

The existence of ${\mathcal{K}}_{\psi}$ doesn’t really follows from §4 because
$\psi$ is not compactly supported. But the choice of $\psi$ is irrelevant and
we can choose $\psi\colon
T^{*}\mathbb{R}^{2n}\xrightarrow[]{}T^{*}\mathbb{R}^{2n}$ given by
$\psi(q_{1},q_{2},p_{1},p_{2})=(q_{1},p_{2},p_{1}+p_{2},q_{1}-q_{2})$ . In
this case we can give a direct construction of ${\mathcal{K}}_{\psi}$ as
follows. Let us define $\Sigma^{+}\subset\mathbb{R}^{4n+1}$ by
$\Sigma^{+}=\\{(q_{1},q_{2},Q_{1},Q_{2},t)\mid
Q_{1}=q_{1},\;t\geq(q_{1}-q_{2})Q_{2}\\}$, whose boundary is the projection of
the lift $\widehat{\Lambda}_{\psi}$ of the graph of $\psi$ to the base. Then
we can check that ${\mathcal{K}}_{\psi}=k_{\Sigma^{+}}$ is the quantization of
$\psi$.

This justifies that both definitions coincide for a Hamiltonian map of
$T^{*}S^{n}$ obtained from a compactly supported map of $T^{*}\mathbb{R}^{n}$.
In the next paragraph we see that the case of a Hamiltonian map of $T^{*}N$
with support in a Darboux chart is reduced to this case.

#### Support in a Darboux chart

Let $S_{\varphi}(q,P,\xi)$ be a generating function quadratic at infinity for
$\Gamma_{\varphi}$ over $\Delta_{{\mathbb{R}}^{2n}}$, that is,

$\left\\{\begin{array}[]{ll}p-P=\frac{\partial S_{\varphi}}{\partial
q}(q,P;\xi)\\\ Q-q=\frac{\partial S_{\varphi}}{\partial P}(q,P;\xi)\\\
\frac{\partial S_{\varphi}}{\partial\xi}(q,P;\xi)=0\\\ \end{array}\right.$

Since $\varphi$ is compactly supported, so is $S_{\varphi}$ in the $q,P$
variables, that is, $S_{\varphi}(q,P;\xi)=B(\xi)$ where $B$ is a non-
degenerate quadratic form, for $|q|+|P|$ large enough. We can then extend
$S_{\varphi}$ to a generating function quadratic at infinity on a bundle over
$S^{2n}$ (see [Vit92]), and $c_{\pm}(\varphi)$ are defined as

$\displaystyle c_{+}(\varphi)$ $\displaystyle=\inf\\{t\mid
T\cup\mu_{S^{2n}}\neq 0\;\text{in}\;H^{*}(S^{t},S^{-\infty})\\},$
$\displaystyle c_{-}(\varphi)$ $\displaystyle=\inf\\{t\mid T\cup 1\neq
0\;\text{in}\;H^{*}(S^{t},S^{-\infty})\\},$

where $T$ is the Thom class associated to the negative bundle of $B$ and
$S^{-\infty}$ means $S^{-c}$ for $c$ large enough. Note that these coincide
with the spectral invariants defined using Floer cohomology (see [Vit95]).

Now if $\varphi$ is defined on $T^{*}N$ but has support in a Darboux chart, it
can be considered either as a compact supported Hamiltonian map on $T^{*}N$ or
on ${\mathbb{R}}^{2n}$ and we can define the spectral invariants in two ways
(using Floer cohomology). By an easy pseudoconvexity argument (see Appendix D)
we can see that these two definitions of $c_{\pm}(\varphi)$ do coincide.

#### Sheaf definition of spectral invariants for a Hamiltonian map

Now we can give another formulation of the definition of $c_{\pm}(\varphi)$.
Let $N$ be a compact manifold and let $\varphi$ be a Hamiltonian map of
$T^{*}N$ with compact support. We consider its homogenized graph
$\widehat{\Lambda}_{\varphi}\subset\overline{T^{*}N}\times T^{*}N\times
T^{*}{\mathbb{R}}$ as in §4. We have associated to $\varphi$ its quantization
${\mathcal{K}}_{\varphi}$, with singular support
$\widehat{\Lambda}_{\varphi}$, and it coincides with ${\mathcal{K}}_{\rm
id}=k_{\Delta_{N}\times[0,+\infty[}$ outside a compact subset of
$N^{2}\times\mathbb{R}$. Now
$\operatorname{\mathcal{H}om}^{\ast}({\mathcal{K}}_{\rm
id},{\mathcal{K}}_{\varphi})$ has similar properties as
$\operatorname{\mathcal{H}om}^{\ast}({\mathcal{F}}_{L_{1}},{\mathcal{F}}_{L_{2}})$,
namely, it is $k_{\Delta_{N}\times\mathbb{R}}[1]$ near
$N^{2}\times\\{-\infty\\}$ and $0$ near $N^{2}\times\\{+\infty\\}$. As in the
case of compact Lagrangians we have
$H^{*}_{N^{2}\times\\{c\\}}(N^{2}\times\mathbb{R};k_{\Delta_{N}\times\mathbb{R}}[1])\simeq
H^{*}(N;k_{N})$ and any class $\alpha\in H^{d}(N;k_{N})$ yields a morphism

$u(\alpha,\varphi,c)\colon{\mathcal{K}}_{\rm
id}\xrightarrow[]{}T_{c*}({\mathcal{K}}_{\varphi})[d],$

for $c$ big enough, which restricts to $\alpha\colon
k_{\Delta_{N}\times\mathbb{R}}\xrightarrow[]{}k_{\Delta_{N}\times\mathbb{R}}[d]$
near $N\times\\{+\infty\\}$ (using ${\rm
Hom}(k_{\Delta_{N}},k_{\Delta_{N}}[d])\simeq H^{d}(N;k_{N})$.

######  Definition 5.6.

For a compactly supported Hamiltonian map $\varphi$ of $T^{*}N$ and $\alpha\in
H^{d}(N;k_{N})$ we set

$c(\alpha,\varphi)=-\inf\\{c;\;\exists u\colon{\mathcal{K}}_{\rm
id}\xrightarrow[]{}T_{t*}({\mathcal{K}}_{\varphi})[d],\;u|_{N^{2}\times]t_{0},+\infty[}=\alpha,\;\text{for
$t_{0}\gg 0$}\\}$

(Here $t_{0}$ is big enough so that over $N^{2}\times]t_{0},+\infty[$ we have
$T_{c*}({\mathcal{K}}_{\varphi})\simeq k_{N\times\mathbb{R}}$.) We also set
$c_{-}(\varphi)=c(1,\varphi)$, $c_{+}(\varphi)=c(\mu_{N},\varphi)$ and
$\gamma(\varphi)=c_{+}(\varphi)-c_{-}(\varphi)$. If $\psi$ is another
compactly supported Hamiltonian map, we set
$\gamma(\varphi,\psi)=\gamma(\varphi\psi^{-1})$.

We have $c_{-}(\varphi)=\inf\\{c(\varphi,\alpha);\;\alpha\in
H^{\star}(N;k_{N})\\}$ and
$c_{+}(\varphi)=\sup\\{c(\varphi,\alpha);\;\alpha\in H^{\star}(N;k_{N})\\}$.

Since $\Gamma(\varphi)$ coincides with the zero section outside a compact set,
we have $c_{-}(\varphi)\leq 0\leq c_{+}(\varphi)$ as in [Vit92] and
$c_{+}(\varphi)=-c_{-}(\varphi^{-1})$.

Note the following analogue of Proposition 5.5

######  Lemma 5.7.

Let $\varphi$ be a compact supported Hamiltonian map. Then there is a morphism
${\mathcal{K}}_{\rm id}\longrightarrow{\mathcal{K}}_{\varphi}$ in
$D^{b}(N\times N\times{\mathbb{R}})$ equal to ${\rm Id}$ at $+\infty$ if and
only if $c_{-}(\varphi)=0$. Analogously there is a morphism
${\mathcal{K}}_{\varphi}\longrightarrow{\mathcal{K}}_{\rm id}$ in
$D^{b}(N\times N\times{\mathbb{R}})$ if and only if $c_{+}(\varphi)=0$.

###  5.3. Defining $\gamma$-coisotropic subsets

For an open set $U$ in the symplectic manifold $(M,\omega)$ we denote by
$\mathfrak{DHam}_{c}(U)$ the set of time one flows of Hamiltonian with compact
support contained in $U$.

######  Definition 5.8 (see definition 6.1 in [Vit22]).

Let $V$ be a set in $(M,\omega)$. We say that $V$ is $\gamma$-coisotropic at
$x\in V$ if, for all pairs of balls $B(x,\eta)\subset B(x,\varepsilon)$ with
$0<\eta<\varepsilon$, there is a $\delta>0$ such that, for all $\varphi$ in
$\mathfrak{DHam}_{c}(B(x,\varepsilon))$ and satisfying $\varphi(V)\cap
B(x,\eta)=\emptyset$, we have $\gamma(\varphi)>\delta$.

We shall say that $V$ is $\gamma$-coisotropic if it is non-empty and
$\gamma$-coisotropic at each $x\in V$.

We refer to [Vit22], in particular section 6, for properties and examples of
$\gamma$-coisotropic subsets. In particular we remind the reader of the
following result from [Vit22]

######  Proposition 5.9.

Let $\varphi$ be a homeomorphism preserving $\gamma$, for example a
$C^{0}$-limit of Hamiltonian maps (usually called Hamiltonian homeomorphisms)
and $V$ a $\gamma$-coisotropic set. Then, the image by $\varphi$ of $V$ is a
$\gamma$-coisotropic set.

Note that the completion of $\mathcal{L}(T^{*}N)$ for $\gamma$ is denoted by
$\widehat{\mathcal{L}}(T^{*}N)$. In [Vit22] the $\gamma$-support of an element
$L$ in $\widehat{\mathcal{L}}(T^{*}N)$ was defined, and it was proved that it
must be a $\gamma$-coisotropic subset.

##  6\. The $\gamma_{g}$-metric on sheaves

We shall define the analogue of the $\gamma$ distance between sheaves. It is
also a sheaf analogue of the barcode distance (see [KS18] or [AI20]) and
relies on the morphism $\tau$ introduced by Tamarkin (see [Tam08] or [GKS,
Prop. 4.8]) that we have already encountered in (5.2).

To define $\tau$ in general we recall that the subcategory $D_{\tau\geq
0}(N\times\mathbb{R})$ of $D(N\times\mathbb{R})$ can be characterized as
follows (see [KS90, Prop. 5.2.3, 3.5.4]). Let $P$ be the endofunctor of
$D(N\times\mathbb{R})$ defined by

$P({\mathcal{F}})=Rs_{*}({\mathcal{F}}\boxtimes k_{[0,+\infty[})$

with $s\colon N\times\mathbb{R}^{2}\xrightarrow[]{}N\times\mathbb{R}$,
$(x,t_{1},t_{2})\mapsto(x,t_{1}+t_{2})$ (this is almost ${\mathcal{F}}\ast
k_{N\times[0,+\infty[}$ up to replacing from $Rs_{!}$ to $Rs_{*}$). Then the
natural morphism $k_{[0,+\infty[}\xrightarrow[]{}k_{\\{0\\}}$ induces a
morphism $P({\mathcal{F}})\xrightarrow[]{}{\mathcal{F}}$ and $D_{\tau\geq
0}(N\times\mathbb{R})$ is formed by the ${\mathcal{F}}$ such that
$P({\mathcal{F}})\xrightarrow[]{}{\mathcal{F}}$ is an isomorphism. Now we have
in general $Rs_{*}({\mathcal{F}}\boxtimes k_{\\{c\\}})\simeq
T_{c*}({\mathcal{F}})$, where $T_{c}(x,t)=(x,t+c)$ and we obtain in the same
way, using the morphism $k_{[0,+\infty[}\xrightarrow[]{}k_{\\{c\\}}$, a
morphism $P({\mathcal{F}})\xrightarrow[]{}T_{c*}({\mathcal{F}})$ for any
$c\geq 0$. In particular we have a morphism of endofunctors defined on
$D_{\tau\geq 0}(N\times\mathbb{R})$ for any $c\geq 0$

$\tau_{0,c}({\mathcal{F}})\colon{\mathcal{F}}\xrightarrow[]{}T_{c*}({\mathcal{F}})$

We shall call this morphism the Tamarkin morphism.

The case of $D_{\tau\geq 0}(N\times\mathbb{R})$ is in fact sufficient for this
paper but we can consider a slightly more general case without modifying the
proofs. Let $g\colon T^{*}N\setminus 0_{N}\xrightarrow[]{}{\mathbb{R}}$ be a
$1$-homogeneous autonomous Hamiltonian function and let $D_{g\geq 0}(N)$ be
the full subcategory of $D(N)$ formed by the ${\mathcal{F}}$ with
$SS^{\bullet}({\mathcal{F}})\subset\\{g\geq 0\\}$. We let
$\varphi^{t}=\varphi_{g}^{t}\colon T^{*}N\setminus
0_{N}\xrightarrow[]{}T^{*}N\setminus 0_{N}$ be the corresponding homogeneous
flow. We set for short $K_{\varphi}^{a}({\mathcal{F}})={\mathcal{F}}\circ
K_{\varphi^{a}}$.

We consider the lifting of the graph of the whole isotopy $\varphi^{t}$ to
Lagrangian in $T^{*}(N\times N\times{\mathbb{R}})$ given by

$\left\\{(q,p,Q_{t}(q,p),P_{t}(q,p),t,g(q,p))\right\\},$

where $\varphi^{t}(q,p)=Q_{t}(q,p),P_{t}(q,p))$ and the associated kernel
$K_{\varphi}\in D(N^{2}\times\mathbb{R})$. Then $K_{\varphi}({\mathcal{F}})\in
D(N\times\mathbb{R})$ is such that
$K_{\varphi}({\mathcal{F}})|_{N\times\\{a\\}}=K_{\varphi}^{a}({\mathcal{F}})$.
Now, if ${\mathcal{F}}\in D_{g\geq 0}(N)$, then $K_{\varphi}({\mathcal{F}})\in
D_{\tau\geq 0}(N\times\mathbb{R})$. Indeed a point of
$SS^{\bullet}(K_{\varphi}({\mathcal{F}}))$ is written $(q,p,Q,P,t,\tau)$ with
$(q,p)\in SS^{\bullet}({\mathcal{F}})$ and $\tau=g(q,p)$. We thus have the
Tamarkin morphism $\tau_{0,c}(K_{\varphi}({\mathcal{F}}))\colon
K_{\varphi}({\mathcal{F}})\xrightarrow[]{}T_{c*}(K_{\varphi}({\mathcal{F}}))$
for $c\geq 0$. Restricting to $N\times\\{a\\}$ and setting $b=a+c$ we obtain

(6.1) $\tau_{a,b}({\mathcal{F}})\colon
K_{\varphi}^{a}({\mathcal{F}})\xrightarrow[]{}K_{\varphi}^{b}({\mathcal{F}}),\qquad\text{for
$a\leq b$.}$

The morphisms $\tau_{a,b}$ are invariant by $K_{\varphi}^{t}$, for any $t$,
and compatible with composition:

(6.2) $\displaystyle K_{\varphi}^{t}(\tau_{a,b}({\mathcal{F}}))$
$\displaystyle=\tau_{a+t,b+t}(K_{\varphi}^{t}({\mathcal{F}})),$
$\displaystyle\tau_{a,c}({\mathcal{F}})$
$\displaystyle=\tau_{b,c}({\mathcal{F}})\circ\tau_{a,b}({\mathcal{F}}),\quad\text{
for }a\leq b\leq c$

######  Examples 6.1.

  1. (1)

Let $N={\mathbb{R}}$ and $g$ be the norm given by the standard metric, so
$g(t,\tau)=|\tau|$. Then $\varphi^{a}(t,\tau)=(t+a\frac{\tau}{|\tau|},\tau)$.
As a result $K_{\varphi^{a}}(k_{[x,y[})=k_{[x+a,y+a[}$ and
$SS(K_{\varphi^{a}})=\left\\{(t,t+a\frac{\tau}{|\tau|},\tau,\tau)\right\\}\subset
T^{*}({\mathbb{R}}^{2})$.

  2. (2)

In the case of $N\times\mathbb{R}$ and $g=\tau$ we recover the case we
introduced in the first paragraph (and originally due to Tamarkin).

  3. (3)

If $g$ is non negative we have $D_{g\geq 0}(N)=D(N)$. Typically $g$ is the
norm given by a Riemannian metric and $\varphi_{g}$ is the normalized geodesic
flow. In this case, denoting by $B(x,r)$ the open ball at $x$ of radius $r$,
we find

$K_{\varphi}^{a}(k_{B(x,r)})\simeq\begin{cases}k_{B(x,r+a)}&\text{for
$-r<a<r_{inj}-r$,}\\\ k_{\overline{B(x,-r-a)}}[-\dim N]\quad&\text{for
$-r_{inj}-r<a\leq-r$,}\end{cases}$

and

$K_{\varphi}^{a}(k_{\overline{B(x,r)}})\simeq\begin{cases}k_{\overline{B(x,r-a)}}&\text{for
$r-r_{inj}\leq a\leq r$,}\\\ k_{{B(x,a-r)}}[\dim N]\quad&\text{for
$r<a<r+r_{inj}$,}\end{cases}$

where $r_{inj}$ is the injectivity radius.

###  6.1. The $\gamma_{g}$-topology

The following definition is the analogue of the spectral metric (in the case
$g=\tau$) (and a sheaf analogue of the barcode distance). It is introduced in
[KS18] or [AI20] (up to a small modification).

######  Definition 6.2.

Let $g\colon T^{*}N\setminus 0_{N}\xrightarrow[]{}{\mathbb{R}}$ be a
$1$-homogeneous autonomous Hamiltonian function and $\varphi=\varphi_{g}^{t}$
its flow. For ${\mathcal{F}},{\mathcal{G}}\in D_{g\geq 0}(N)$, we let
$\gamma_{g}({\mathcal{F}},{\mathcal{G}})\in[0,\infty]$ be the infimum of the
$c\geq 0$ for which there exist $a,b\geq 0$ such that $a+b=c$ and such that
there are morphisms

$u\colon{\mathcal{F}}\xrightarrow[]{}K_{\varphi}^{a}({\mathcal{G}}),\qquad
v\colon{\mathcal{G}}\xrightarrow[]{}K_{\varphi}^{b}({\mathcal{F}})$

satisfying $K_{\varphi}^{a}(v)\circ u=\tau_{0,a+b}({\mathcal{F}})$ and
$K_{\varphi}^{b}(u)\circ v=\tau_{0,a+b}({\mathcal{G}})$.

The authors thank N. Vichery for the next remark.

######  Remark 6.3.

The case $D_{g\geq 0}(N)$ is not really more general than the case
$D_{\tau\geq 0}(N\times\mathbb{R})$. Indeed the functor $K_{\varphi}(-)\colon
D(N)\xrightarrow[]{}D(N\times\mathbb{R})$ is fully faithful and we have
$T_{a*}(K_{\varphi}(-))\simeq K_{\varphi}(K_{\varphi}^{a}(-))$. Hence, for
${\mathcal{F}},{\mathcal{G}}\in D_{g\geq 0}(N)$ we have
$\gamma_{g}({\mathcal{F}},{\mathcal{G}})=\gamma_{\tau}({\mathcal{K}}_{\varphi}({\mathcal{F}}),{\mathcal{K}}_{\varphi}({\mathcal{G}}))$.

The definition in [KS18] requires $a=b$ which leads to a bigger distance, say
$\gamma^{\prime}_{g}$, with $\gamma_{g}\leq\gamma^{\prime}_{g}\leq
2\gamma_{g}$. The definition in [AI20] only asks $K_{\varphi}^{a}(v)\circ
u=\tau_{0,a+b}({\mathcal{F}})$ and $K_{\varphi}^{b}(u^{\prime})\circ
v^{\prime}=\tau_{0,a+b}({\mathcal{G}})$, for two possibly different morphisms
$u^{\prime}$, $v^{\prime}$. This leads to a smaller distance but we don’t know
if it is equivalent to $\gamma_{g}$. Lemma 6.8 below says that $\gamma_{g}$ is
equivalent to $\inf\\{\gamma_{g}({\mathcal{C}}(u),0)$;
$u\colon{\mathcal{F}}\xrightarrow[]{}{\mathcal{G}}\\}$, where
${\mathcal{C}}(u)$ denotes the cone of $u$.

######  Lemma 6.4.

For ${\mathcal{F}},{\mathcal{G}},\mathcal{H}\in D_{g\geq 0}(N)$ we have

  1. (1)

$\gamma_{g}({\mathcal{F}},{\mathcal{G}})=\gamma_{g}(K_{\varphi}^{t}({\mathcal{F}}),K_{\varphi}^{t}({\mathcal{G}}))=\gamma_{g}({\mathcal{G}},{\mathcal{F}})$.

  2. (2)

$\gamma_{g}({\mathcal{F}},\mathcal{H})\leq\gamma_{g}({\mathcal{F}},{\mathcal{G}})+\gamma_{g}({\mathcal{G}},\mathcal{H})$.

  3. (3)

$\gamma_{g}(0,{\mathcal{F}})=\inf\\{t\geq 0$;
$\tau_{0,t}({\mathcal{F}})=0\\}$.

  4. (4)

If ${\mathcal{F}}$ satisfies $\gamma_{g}({\mathcal{F}},0)=0$, then
${\mathcal{F}}\simeq 0$.

  5. (5)

For any distinguished triangle
${\mathcal{F}}\xrightarrow[]{u}{\mathcal{G}}\xrightarrow[]{v}\mathcal{H}\smash{\xrightarrow[]{+1}}$,
we have

$\gamma_{g}(0,{\mathcal{G}})\leq\gamma_{g}(0,{\mathcal{F}})+\gamma_{g}(0,\mathcal{H})$

  6. (6)

For composable morphisms $u$, $v$ with cones ${\mathcal{C}}(u)$,
${\mathcal{C}}(v)$, ${\mathcal{C}}(v\circ u)$, we have
$\gamma_{g}(0,{\mathcal{C}}(v\circ
u))\leq\gamma_{g}(0,{\mathcal{C}}(u))+\gamma_{g}(0,{\mathcal{C}}(v))$.

###### Proof.

(1) follows from (6.2) and (3) follows from the definition of $\gamma_{g}$.
The triangular inequality (2) is proved in [AI20, Prop. 4.11].

Let us prove (4). By Remark 6.3 we can assume that $N=M\times\mathbb{R}$ and
$g=\tau$. Let $x_{0}\in M$ be given and let
$i\colon\\{x_{0}\\}\times\mathbb{R}\xrightarrow[]{}M\times\mathbb{R}$ be the
inclusion. We have
$\tau_{a,b}(i^{-1}{\mathcal{F}})=i^{-1}(\tau_{a,b}({\mathcal{F}}))$ and it
follows that
$\gamma_{g}(i^{-1}{\mathcal{F}},i^{-1}{\mathcal{G}})\leq\gamma_{g}({\mathcal{F}},{\mathcal{G}})$.
If ${\mathcal{F}}\not=0$, there exists $x_{0}$ such that
$i^{-1}({\mathcal{F}})\not=0$. Hence we can assume that $N=\mathbb{R}$. We can
also take $g=|\tau|$ (it is differentiable on $T^{*}\mathbb{R}\setminus
0_{\mathbb{R}}$) instead of $g=\tau$. The corresponding $\tau$ morphisms, say
$\tau^{g}_{a,b}$, coincide with $\tau_{a,b}$ on $D_{\tau\geq 0}(\mathbb{R})$
but they are defined on $D(\mathbb{R})$. Up to shifting ${\mathcal{F}}$ in
degrees and translating, we can assume that $H^{0}{\mathcal{F}}_{0}\not=0$.
Hence there exists $u\in
H^{0}(\mathopen{]}-\varepsilon,\varepsilon[;{\mathcal{F}})$ with
$u_{0}\not=0$. Interpreting $u$ as a morphism and chosing a map
$v:{\mathcal{F}}_{0}\longrightarrow k$ non-vanishing on the germ defined by
$u$, we get that the composition of $v$ and $u$;
$k_{]-\varepsilon,\varepsilon[}\xrightarrow[]{u}{\mathcal{F}}\xrightarrow[]{v}k_{\\{0\\}}$
is the canonical morphism. Now the morphisms $\tau^{g}_{0,\delta}(-)$, for
$0<\delta<\varepsilon$ give the commutative diagram (using Example 6.1 (3) )
where the vertical arrows are given by $\tau^{g}_{0,\delta}$

$\textstyle{k_{]-\varepsilon,\varepsilon[}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\textstyle{{\mathcal{F}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{v}$$\textstyle{k_{\\{0\\}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{w}$$\textstyle{k_{]-\varepsilon+\delta,\varepsilon-\delta[}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{T_{\delta*}({\mathcal{F}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{k_{]-\delta,\delta[}[1]}$

We have $w\circ v\circ u\not=0$ and we deduce that
$\tau^{g}_{0,\delta}({\mathcal{F}})\not=0$, as required.

Let us prove (5) (a similar but more detailed argument will be given in the
proof of Lemma 6.8-(i)). If $\tau_{0,a}({\mathcal{F}})=0$, then
$\tau_{0,a}({\mathcal{G}})\circ
u=K_{\varphi}^{a}(u)\circ\tau_{0,a}({\mathcal{F}})=0$ and
$\tau_{0,a}({\mathcal{G}})$ factorizes through $v$. In the same way, if
$\tau_{0,b}(\mathcal{H})=0$, then $\tau_{0,b}({\mathcal{G}})$ factorizes
through $K_{\varphi}^{b}(v)$. Hence $\tau_{0,a+b}({\mathcal{G}})$ factorizes
through $K_{\varphi}^{b}(v)\circ K_{\varphi}^{b}(u)=0$. We conclude with (3).

We deduce (6) from (5) since, by the octaedron axiom, the three cones appear
in a distinguished triangle. ∎

If $N$ is real analytic and ${\mathcal{F}},{\mathcal{G}}$ are constructible
and satisfy $\gamma_{g}({\mathcal{F}},{\mathcal{G}})=0$, then
${\mathcal{F}}\simeq{\mathcal{G}}$ (see [PSW] and also Proposition B.7 for
limits of constructible sheaves). However $\gamma_{g}$ is only a pseudo-metric
as shown by Berkouk and Ginot (see [BG18], proposition 6.9) and the following
example.

######  Example 6.5.

We define two sheaves on the real line
${\mathcal{F}}=\bigoplus_{x\in\mathbb{Q}}k_{[x,\infty[}$ and
${\mathcal{G}}=\bigoplus_{x\in\mathbb{Q}^{\times}}k_{[x,\infty[}$, where
$\mathbb{Q}^{\times}=\mathbb{Q}\setminus\\{0\\}$. We remark that
$K_{\varphi}^{t}({\mathcal{F}})\simeq T_{t*}({\mathcal{F}})$, where
$T_{t}(x)=x+t$. For a given $\varepsilon>0$ we choose a bijection
$f\colon\mathbb{Q}\xrightarrow[]{}\mathbb{Q}^{\times}$ such that $x\leq
f(x)\leq x+\varepsilon$, for all $x\in\mathbb{Q}$, and we define
$u=\bigoplus_{x\in\mathbb{Q}}u_{x}\colon{\mathcal{F}}\xrightarrow[]{}{\mathcal{G}}$,
$v=\bigoplus_{x\in\mathbb{Q}^{\times}}v_{x}\colon{\mathcal{G}}\xrightarrow[]{}T_{\varepsilon*}{\mathcal{F}}$
where $u_{x}\colon k_{[x,\infty[}\xrightarrow[]{}k_{[f(x),\infty[}$ and
$v_{x}\colon k_{[x,\infty[}\xrightarrow[]{}k_{[f^{-1}(x)+\varepsilon,\infty[}$
are the natural morphisms. Using $u,v$ we see that
$\gamma_{g}({\mathcal{F}},{\mathcal{G}})\leq\varepsilon$. Hence
$\gamma_{g}({\mathcal{F}},{\mathcal{G}})=0$.

######  Definition 6.6.

We denote by $\gamma_{g}$ the topology associated to the pseudo-distance
$\gamma_{g}$. In other words a sequence $({\mathcal{F}}_{j})_{j\geq 1}$ in
$D(N)$ $\gamma_{g}$-converges to ${\mathcal{F}}$ if and only if
$\lim_{j}\gamma_{g}({\mathcal{F}}_{j},{\mathcal{F}})=0$.

######  Remark 6.7.

In view of the Lemma 6.8, a sequence $({\mathcal{F}}_{n})_{n\in\mathbb{N}}$
converges to ${\mathcal{F}}$ if and only if there exist a sequence of positive
numbers $(\varepsilon_{n})_{n\in\mathbb{N}}$ converging to $0$ and morphisms
$u_{n}\colon{\mathcal{F}}_{n}\xrightarrow[]{}K_{\varphi}^{\varepsilon_{n}}({\mathcal{F}})$
such that $\gamma(0,{\mathcal{C}}(u_{n}))$ converges to $0$.

Part (2) in the next lemma is analogue to [AI20, Lem. 4.14].

######  Lemma 6.8.

Let $u\colon{\mathcal{F}}\xrightarrow[]{}{\mathcal{F}}^{\prime}$ be a morphism
in $D(N)$ and let ${\mathcal{C}}$ be the cone of $u$.

  1. (1)

If there exists
$v\colon{\mathcal{F}}^{\prime}\xrightarrow[]{}K_{\varphi}^{\varepsilon}({\mathcal{F}})$
such that $v\circ u=\tau_{0,\varepsilon}$ and
$K_{\varphi}^{\varepsilon}(u)\circ v=\tau_{0,2\varepsilon}$, then
$\gamma({\mathcal{C}},0)\leq 2\varepsilon$.

  2. (2)

If $\gamma({\mathcal{C}},0)<\varepsilon$, then there exists
$v\colon{\mathcal{F}}^{\prime}\xrightarrow[]{}K_{\varphi}^{2\varepsilon}({\mathcal{F}})$
such that $v\circ u=\tau_{0,2\varepsilon}$ and
$K_{\varphi}^{2\varepsilon}(u)\circ v=\tau_{0,4\varepsilon}$.

###### Proof.

(1) We consider the morphism of triangles given by the functorial morphism
(6.1),

$\textstyle{{\mathcal{F}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\textstyle{{\mathcal{F}}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{v}$$\scriptstyle{f}$$\textstyle{{\mathcal{C}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\scriptstyle{g}$$\textstyle{{\mathcal{F}}[1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{\varphi}^{\varepsilon}({\mathcal{F}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{\varphi}^{\varepsilon}({\mathcal{F}}^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{\varphi}^{\varepsilon}({\mathcal{C}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{\varphi}^{\varepsilon}({\mathcal{F}})[1]}$

Here all small triangles commute (which defines $f$ and $g$). We have
$f=K_{\varphi}^{\varepsilon}(\alpha)\circ\tau_{0,\varepsilon}({\mathcal{F}}^{\prime})=K_{\varphi}^{\varepsilon}(\alpha\circ
u)\circ v=0$. In the same way $g$ also vanishes. Since
$\tau_{0,\varepsilon}({\mathcal{C}})\circ\alpha=f$ vanishes,
$\tau_{0,\varepsilon}({\mathcal{C}})$ factorizes through $\beta$ as
$\tau_{0,\varepsilon}({\mathcal{C}})=h\circ\beta$. Hence
$\tau_{0,2\varepsilon}({\mathcal{C}})=K_{\varphi}^{\varepsilon}(\tau_{0,\varepsilon}({\mathcal{C}}))\circ\tau_{0,\varepsilon}({\mathcal{C}})=K_{\varphi}^{\varepsilon}(h)\circ
K_{\varphi}^{\varepsilon}(\beta)\circ\tau_{0,\varepsilon}({\mathcal{C}})=K_{\varphi}^{\varepsilon}(h)\circ
g$ vanishes, which means $\gamma({\mathcal{C}},0)\leq 2\varepsilon$.

(2) Since $\tau_{0,\varepsilon}({\mathcal{C}})=0$, both morphisms $f$ and $g$
vanish. Since
$K_{\varphi}^{\varepsilon}(\alpha)\circ\tau_{0,\varepsilon}({\mathcal{F}}^{\prime})=f$
vanishes, $\tau_{0,\varepsilon}({\mathcal{F}}^{\prime})$ factorizes as
$\tau_{0,\varepsilon}({\mathcal{F}}^{\prime})=K_{\varphi}^{\varepsilon}(u)\circ
w$ for some $w$.

In the same way, the vanishing of $g$ gives a factorization
$\tau_{0,\varepsilon}({\mathcal{F}})=w^{\prime}\circ u$ but we cannot say that
$w=w^{\prime}$.

We set
$v=K_{\varphi}^{\varepsilon}(w^{\prime})\circ\tau_{0,\varepsilon}({\mathcal{F}}^{\prime})\colon{\mathcal{F}}^{\prime}\xrightarrow[]{}K_{\varphi}^{2\varepsilon}({\mathcal{F}})$.
In the following computation we omit abusively to write
$K_{\varphi}^{\varepsilon}$ and write $\tau$ for whatever $\tau_{-,-}(-)$. A
diagram chase on the diagram below gives $v\circ u=w^{\prime}\circ\tau\circ
u=w^{\prime}\circ u\circ\tau=\tau\circ\tau=\tau$ and $u\circ v=u\circ
w^{\prime}\circ\tau=u\circ w^{\prime}\circ u\circ w=u\circ\tau\circ
w=\tau\circ u\circ w=\tau\circ\tau=\tau$, as required.

$\textstyle{{\mathcal{F}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\textstyle{{\mathcal{F}}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{w}$$\scriptstyle{\circlearrowleft}$$\textstyle{K_{\varphi}^{\varepsilon}({\mathcal{F}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\textstyle{K_{\varphi}^{\varepsilon}({\mathcal{F}}^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{w^{\prime}}$$\scriptstyle{\circlearrowright}$$\textstyle{K_{\varphi}^{2\varepsilon}({\mathcal{F}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\textstyle{K_{\varphi}^{2\varepsilon}({\mathcal{F}}^{\prime})}$

∎

###  6.2. Link with spectral invariants

In this paragraph we work on $N\times\mathbb{R}$ and choose $g=\tau$. So we
work with the category $D_{\tau\geq 0}(N\times\mathbb{R})$ as in §5. We prove
Propositions 6.11 and 6.13 which explain the relations between the
$\gamma_{\tau}$-distance and the spectral invariants.

We first give a useful variation on the Morse theorem for sheaves. The
following result is mentioned in [Nad16, §6.1].

######  Proposition 6.9.

Let $M$ be a manifold and $I$ an open interval of $\mathbb{R}$. Let
${\mathcal{F}}\in D(M),{\mathcal{G}}\in D(M\times I)$ and set
${\mathcal{G}}_{t}={\mathcal{G}}|_{M\times\\{t\\}}$ for $t\in I$. We assume

  1. (1)

there exists a compact set $C$ of $M$ such that ${\mathcal{G}}|_{(M\setminus
C)\times I}$ is of the form ${\mathcal{G}}^{\prime}\boxtimes k_{I}$ for some
${\mathcal{G}}^{\prime}\in D(M\setminus C)$,

  2. (2)

$SS^{\bullet}({\mathcal{G}})\cap(0_{M}\times T^{*}I)=\emptyset$,

  3. (3)

$(SS^{\bullet}({\mathcal{F}})\times T^{*}I)\cap
SS^{\bullet}({\mathcal{G}})=\emptyset$.

Then $\operatorname{RHom}({\mathcal{F}},{\mathcal{G}}_{t})$ does not depend on
$t\in I$.

###### Proof.

We let $i_{t}\colon M\times\\{t\\}\xrightarrow[]{}M\times I$, $t\in I$, be the
inclusion. We set
$\mathcal{H}=\operatorname{R\mathcal{H}om}({\mathcal{F}}\boxtimes
k_{I},{\mathcal{G}})$. By [KS90, Prop. 5.4.14], condition (3) gives
$SS(\mathcal{H})\subset\Lambda:=(SS({\mathcal{F}})\times
0_{I})^{a}+SS({\mathcal{G}})$. We can see moreover that $\Lambda$ is also non-
characteristic for all $i_{t}$ (that is, $\Lambda\cap(0_{M}\times
T^{*}I)\subset 0_{M\times I}$). Using condition (2)
$i_{t}^{-1}{\mathcal{G}}\simeq i_{t}^{!}{\mathcal{G}}[1]$ and
$i_{t}^{-1}\mathcal{H}\simeq i_{t}^{!}\mathcal{H}[1]$ by [KS90, Prop. 5.4.13]
and we deduce
$\operatorname{R\mathcal{H}om}({\mathcal{F}},{\mathcal{G}}_{t})\simeq
i_{t}^{-1}\mathcal{H}$ by the adjunction formula
$i_{t}^{!}\operatorname{R\mathcal{H}om}(-,-)\simeq\operatorname{R\mathcal{H}om}(i_{t}^{-1}(-),i_{t}^{!}(-))$.
Hence
$\operatorname{RHom}({\mathcal{F}},{\mathcal{G}}_{t})\simeq\operatorname{R\Gamma}(M;i_{t}^{-1}\mathcal{H})$.
By the condition (1) $\mathcal{H}|_{(M\setminus C)\times I}$ is of the form
$\mathcal{H}^{\prime}\boxtimes k_{I}$, for some $\mathcal{H}^{\prime}\in
D(M\setminus C)$, and we can use the base change formula and the bound for the
singular support of a direct image. Hence
$\operatorname{R\Gamma}(M;i_{t}^{-1}\mathcal{H})\simeq(Rq_{*}(\mathcal{H}))_{t}$,
where $q\colon M\times I\xrightarrow[]{}I$ is the projection and it is enough
to check that $Rq_{*}(\mathcal{H})$ is constant. Since
$\Lambda\cap(0_{M}\times T^{*}I)\subset 0_{M\times I}$, we have
$SS(Rq_{*}(\mathcal{H}))\subset 0_{I}$ and the result follows. ∎

######  Corollary 6.10.

Let ${\mathcal{F}}\in D_{\tau\geq 0}(N\times\mathbb{R})$ and $t_{0}$ satisfy
(5.3). We assume that $SS({\mathcal{F}})\cap SS(T_{c*}({\mathcal{F}}))\subset
0_{N\times\mathbb{R}}$ for any $c\not=0$. Then the restriction map

$\operatorname{RHom}({\mathcal{F}},T_{c*}({\mathcal{F}}))\xrightarrow[]{}\operatorname{RHom}({\mathcal{F}}|_{U},T_{c*}({\mathcal{F}})|_{U})\simeq\operatorname{RHom}(k_{U},k_{U})\simeq\operatorname{R\Gamma}(N;k_{N})$

is an isomorphism for any $c>0$, where $U=N\times]c+t_{0},+\infty[$. In
particular, if $u\colon{\mathcal{F}}\xrightarrow[]{}T_{c*}({\mathcal{F}})$ is
${\rm id}$ near $+\infty$, then $u=\tau_{0,c}({\mathcal{F}})$.

When $SS^{\bullet}({\mathcal{F}})$ is the cone over a Legendrian, the
hypothesis means that there is no Reeb chord. In particular the hypothesis is
satisfied for the sheaves ${\mathcal{F}}_{L}$, $L\in\mathcal{L}(T^{*}N)$.

###### Proof.

We are going to apply Proposition 6.9 with $M=N\times\mathbb{R}$, $I=]0,d]$
for $d>0$ and ${\mathcal{G}}=\delta^{-1}({\mathcal{F}})$, where $\delta\colon
N\times\mathbb{R}^{2}\xrightarrow[]{}N\times\mathbb{R}$ is the map

$(x,t_{1},t_{2})\mapsto(x,t_{1}-t_{2})$

so that ${\mathcal{G}}|_{N\times\mathbb{R}\times\\{c\\}}\simeq
T_{c*}({\mathcal{F}})$.

For $a<b$, ${\mathcal{G}}$ is constant on
$N\times]-\infty,a-t_{0}[\times[a,b]$ and
$N\times]b+t_{0},+\infty[\times[a,b]$. Hence ${\mathcal{G}}$ satisfies (1) in
Proposition 6.9. It is not difficult to check (2) since its singular support
is contained in the conormal of $t_{1}=t_{2}$ that is $\tau_{1}+\tau_{2}=0$.
For (3) it follows from the hypothesis on $SS({\mathcal{F}})$. The proposition
gives
$\operatorname{RHom}({\mathcal{F}},T_{c*}({\mathcal{F}}))\simeq\operatorname{RHom}({\mathcal{F}},T_{d*}({\mathcal{F}}))$,
for any $0<c\leq d$. ∎

######  Proposition 6.11.

Let $L_{1},L_{2}\in\mathcal{L}(T^{*}N)$. Then

$\gamma_{\tau}({\mathcal{F}}_{L_{1}},{\mathcal{F}}_{L_{2}})=\begin{cases}c_{+}(L_{1},L_{2})&\text{if
$c_{-}(L_{1},L_{2})>0$,}\\\ -c_{-}(L_{1},L_{2})&\text{if
$c_{+}(L_{1},L_{2})<0$,}\\\
c_{+}(L_{1},L_{2})-c_{-}(L_{1},L_{2})\quad&\text{else.}\end{cases}$

###### Proof.

We recall that ${\mathcal{F}}_{L_{i}}\simeq k_{N\times\mathbb{R}}$ near
$+\infty$ and that $-c_{-}(L_{1},L_{2})$ is the infimum on the $a$ such that
there exists
$u\colon{\mathcal{F}}_{L_{1}}\xrightarrow[]{}T_{a*}({\mathcal{F}}_{L_{2}})$
restricting to ${\rm id}_{k_{N\times\mathbb{R}}}$ near $+\infty$. Similarly
$c_{+}(L_{1},L_{2})$ is the infimum on the $b$ such that there exists
$v\colon{\mathcal{F}}_{L_{2}}\xrightarrow[]{}T_{b*}({\mathcal{F}}_{L_{1}})$
restricting to ${\rm id}_{k_{N\times\mathbb{R}}}$ near $+\infty$. We then have
$T_{a*}(v)\circ u=\tau_{0,a+b}({\mathcal{F}}_{L_{1}})$ and $T_{b*}(u)\circ
v=\tau_{0,a+b}({\mathcal{F}}_{L_{2}})$ by Corollary 6.10. Hence $u$, $v$
satisfy Definition 6.2, except that here $a,b$ may be negative. Assuming
moreover $a,b\geq 0$ we find that
$\gamma_{\tau}({\mathcal{F}}_{L_{1}},{\mathcal{F}}_{L_{2}})$ is less than the
right hand side of the equality in the statement.

Conversely, if $\gamma_{\tau}({\mathcal{F}}_{L_{1}},{\mathcal{F}}_{L_{2}})<c$,
there exists $a,b,u,v$ as above with $a+b=c$, $a,b\geq 0$ but $u$, $v$ only
satisfy the condition on the composition. Near $+\infty$ we only know that
they are $u=\lambda_{u}{\rm id}_{k_{N\times\mathbb{R}}}$, $v=\lambda_{v}{\rm
id}_{k_{N\times\mathbb{R}}}$ (indeed ${\rm
Hom}(k_{N\times\mathbb{R}},k_{N\times\mathbb{R}})\simeq k$). Since
$T_{a*}(v)\circ u=\tau_{0,a+b}({\mathcal{F}}_{L_{1}})$, we have
$\lambda_{u}\lambda_{v}=1$. Hence $\lambda_{u}$, $\lambda_{v}$ are not zero
and, replacing $u$, $v$ by a non zero multiple, we can assume that $u=v={\rm
id}_{k_{N\times\mathbb{R}}}$ near $+\infty$. We deduce
$a\geq\max\\{-c_{-}(L_{1},L_{2}),0\\}$, $b\geq\max\\{c_{+}(L_{1},L_{2}),0\\}$.
∎

######  Remark 6.12.

Note that in [Vit22] the metric on $\mathcal{L}(T^{*}N)$ is defined by
$c(L_{1},L_{2})=|c_{+}(L_{1},L_{2})|+|c_{-}(L_{1},L_{2})|$ as . The map
$L\mapsto{\mathcal{F}}_{L}$ is then a bi-Lipschitz embedding. Indeed if
$c_{-}(L_{1},L_{2})>0$ then
$\gamma_{\tau}({\mathcal{F}}_{L_{1}},{\mathcal{F}}_{L_{2}})=c_{+}(L_{1},L_{2})$
and since $c_{-}(L_{1},L_{2})\leq c_{+}(L_{1},L_{2})$, we have

$\gamma_{\tau}({\mathcal{F}}_{L_{1}},{\mathcal{F}}_{L_{2}})\leq
c(L_{1},L_{2})=c_{+}(L_{1},L_{2})+c_{-}(L_{1},L_{2})\leq
2\gamma_{\tau}({\mathcal{F}}_{L_{1}},{\mathcal{F}}_{L_{2}})$

If $c_{+}(L_{1},L_{2})<0$ the same inequality holds by exchanging
$L_{1},L_{2}$. Finally if $c_{-}(L_{1},L_{2})<0<c_{+}(L_{1},L_{2})$ we have
$\gamma_{\tau}({\mathcal{F}}_{L_{1}},{\mathcal{F}}_{L_{2}})=c_{+}(L_{1},L_{2})-c_{-}(L_{1},L_{2})$
hence

$\gamma_{\tau}({\mathcal{F}}_{L_{1}},{\mathcal{F}}_{L_{2}})=c(L_{1},L_{2})$

So, in all three cases we have

$\gamma_{\tau}({\mathcal{F}}_{L_{1}},{\mathcal{F}}_{L_{2}})\leq
c(L_{1},L_{2})\leq
2\gamma_{\tau}({\mathcal{F}}_{L_{1}},{\mathcal{F}}_{L_{2}})$

The proof of Proposition 6.11 works exactly the same way for Hamiltonian maps,
replacing the condition “${\mathcal{F}}_{L_{i}}\simeq k_{N\times\mathbb{R}}$
near $+\infty$” by “${\mathcal{K}}_{\varphi}\simeq
k_{\Delta_{N}\times[0,+\infty[}$ near $+\infty$”. Since we know that
$c_{-}(\varphi)\leq 0\leq c_{+}(\varphi)$ by [Vit92] we obtain the following
statement:

######  Proposition 6.13.

Let $\varphi_{1}$, $\varphi_{2}$ be compactly supported Hamiltonian maps of
$T^{*}N$. Then
$\gamma_{\tau}({\mathcal{K}}_{\varphi_{1}},{\mathcal{K}}_{\varphi_{2}})=\gamma(\varphi_{1},\varphi_{2})$.

We end this section by the useful result that $\varoast$ is Lipschitz for the
$\gamma_{\tau}$-distance.

######  Lemma 6.14.

Let $M$, $N$, $P$ be manifolds and ${\mathcal{K}}_{1}\in D_{\tau\geq
0}(M\times N\times\mathbb{R})$, ${\mathcal{K}}_{2}\in D_{\tau\geq 0}(N\times
P\times\mathbb{R})$. Then, for any $a\in\mathbb{R}$, we have natural
isomorphisms $T_{a*}({\mathcal{K}}_{1}\varoast{\mathcal{K}}_{2})\simeq
T_{a*}({\mathcal{K}}_{1})\varoast{\mathcal{K}}_{2}\simeq{\mathcal{K}}_{1}\varoast
T_{a*}({\mathcal{K}}_{2})$ and, through these isomorphisms, we have the
equalities, for any $a\leq b$,

$\tau_{a,b}({\mathcal{K}}_{1}\varoast{\mathcal{K}}_{2})=\tau_{a,b}({\mathcal{K}}_{1})\varoast{\mathcal{K}}_{2}={\mathcal{K}}_{1}\varoast\tau_{a,b}({\mathcal{K}}_{2})$

######  Remark 6.15.

The notation $\tau_{a,b}({\mathcal{K}}_{1})\varoast{\mathcal{K}}_{2}$ means
that we apply the functor $\varoast{\mathcal{K}}_{2}$ to the morphism
$\tau_{a,b}({\mathcal{K}}_{1})$. This means that
$(\tau_{a,b}({\mathcal{K}}_{1})\varoast{\mathcal{K}}_{2})$ is a morphism

$\tau_{a,b}({\mathcal{K}}_{1})\varoast{\mathcal{K}}_{2}:{T_{a}}_{*}({\mathcal{K}}_{1})\varoast{\mathcal{K}}_{2}\longrightarrow{T_{b}}_{*}({\mathcal{K}}_{1})\varoast{\mathcal{K}}_{2}$

###### Proof.

In the proof the manifolds $M,N,P$ play an auxiliary role and we can consider
that our sheaves live on $\mathbb{R}$ to simplify the notations. Hence
${\mathcal{K}}_{1}\varoast{\mathcal{K}}_{2}=Rs_{!}({\mathcal{K}}_{1}\boxtimes{\mathcal{K}}_{2})$
where $s\colon\mathbb{R}^{2}\xrightarrow[]{}\mathbb{R}$ is the sum. The
isomorphisms then follow from the equalities $T_{a}\circ
s=s\circ(T_{a}\times{\rm id}_{\mathbb{R}})=s\circ({\rm id}_{\mathbb{R}}\times
T_{a})$. However to prove the equalities of morphisms we need to be more
precise.

Let $(t_{1},t_{2},\tau_{1},\tau_{2})$ be the coordinates on
$T^{*}\mathbb{R}^{2}$. Then the singular support of
${\mathcal{K}}_{1}\boxtimes{\mathcal{K}}_{2}$ is contained in $\\{\tau_{1}\geq
0\\}\times\\{\tau_{2}\geq 0\\}$ and, setting $g_{u}=u\tau_{1}+(1-u)\tau_{2}$,
${\mathcal{K}}_{1}\boxtimes{\mathcal{K}}_{2}$ belongs to $D_{g_{u}\geq
0}(\mathbb{R}^{2})$ for any $u\in[0,1]$. Let us write $\tau_{a,b}^{u}$ for the
morphism $\tau_{a,b}$ induced by $g_{u}$. Then $\tau_{a,b}^{1}$ gives
$\tau_{a,b}({\mathcal{K}}_{1})\varoast{\mathcal{K}}_{2}$ by appyling $Rs_{!}$
and $\tau_{a,b}^{0}$ gives
${\mathcal{K}}_{1}\varoast\tau_{a,b}({\mathcal{K}}_{2})$. It remains to see
that $\tau_{a,b}^{u}$ gives the same morphism for all $u$. We define
${\mathcal{K}}={\mathcal{K}}_{1}\boxtimes{\mathcal{K}}_{2}\boxtimes
k_{[0,1]}\in D(\mathbb{R}^{2}\times[0,1])$ and $g=u\tau_{1}+(1-u)\tau_{2}$.
Then $K\in D_{g\geq 0}(\mathbb{R}^{2}\times[0,1])$ (here we are cheating a
little bit because the singular support is not defined on a manifold with
boundary – however we can replace $[0,1]$ by
$\mathopen{]}-\varepsilon,1+\varepsilon[$ and $u$ by a smooth function with
values in $[0,1]$ which is $0$ on $\mathopen{]}-\varepsilon,0]$ and $1$ on
$[1,1+\varepsilon[$). We thus obtain a morphism $\tau_{a,b}^{g}\colon
T^{\prime}_{a*}({\mathcal{K}})\xrightarrow[]{}T^{\prime}_{b*}({\mathcal{K}})$
where $T^{\prime}_{c}(t_{1},t_{2},u)=(t_{1}+uc,t_{2}+(1-u)c,u)$. Let
$s^{\prime}\colon\mathbb{R}^{2}\times[0,1]\xrightarrow[]{}\mathbb{R}\times[0,1]$
be the sum of the first two variables. Then $s^{\prime}\circ
T^{\prime}_{c}=T_{c}\times{\rm id}_{[0,1]}$ and $\tau_{a,b}^{g}$ induces
$\tau^{\prime}_{a,b}\colon(T_{a*}({\mathcal{K}}_{1}\varoast{\mathcal{K}}_{2}))\boxtimes
k_{[0,1]}\xrightarrow[]{}(T_{b*}({\mathcal{K}}_{1}\varoast{\mathcal{K}}_{2}))\boxtimes
k_{[0,1]}$ which restricts to $\tau_{a,b}^{s}$ along
$\mathbb{R}\times\\{s\\}$. Now restricting along $\mathbb{R}\times\\{s\\}$
always induces the same isomorphism ${\rm Hom}({\mathcal{F}}\boxtimes
k_{[0,1]},{\mathcal{G}}\boxtimes
k_{[0,1]})\xrightarrow[]{{\raisebox{-2.58334pt}[0.0pt][-2.58334pt]{$\mspace{1.0mu}\sim\mspace{2.0mu}$}}}{\rm
Hom}({\mathcal{F}},{\mathcal{G}})$, for all ${\mathcal{F}}$, ${\mathcal{G}}$,
namely, the inverse of the isomorphism induced by the pull-back along the
projection to $[0,1]$. This concludes the proof. ∎

We deduce an analogue of [PS21], theorem 2.4.1, in our situation.

######  Lemma 6.16.

Let ${\mathcal{K}}_{1},{\mathcal{K}}_{2}\in D_{\tau\geq 0}(M\times
N\times\mathbb{R})$ and ${\mathcal{F}}\in D_{\tau\geq 0}(M\times\mathbb{R})$.
Then ${\mathcal{K}}_{j}\varoast{\mathcal{F}}\in D_{\tau\geq
0}(N\times\mathbb{R})$ and we have

$\gamma_{\tau}({\mathcal{K}}_{1}^{\varoast}({\mathcal{F}}),{\mathcal{K}}_{2}^{\varoast}({\mathcal{F}}))\leq\gamma_{\tau}({\mathcal{K}}_{1},{\mathcal{K}}_{2})$

###### Proof.

We have $\gamma_{\tau}({\mathcal{K}}_{1},{\mathcal{K}}_{2})<c$, for a given
$c$, if and only if there exist $a,b\geq 0$ with $a+b=c$ and morphisms
$u\colon{\mathcal{K}}_{1}\xrightarrow[]{}T_{a*}({\mathcal{K}}_{2})$,
$v\colon{\mathcal{K}}_{2}\xrightarrow[]{}T_{b*}({\mathcal{K}}_{1})$ satisfying
the conditions in Definition 6.2. The morphisms induce $u\circ{\rm
id}_{\mathcal{F}}$, $v\circ{\rm id}_{\mathcal{F}}$ which also satisfies the
conditions of the definition by Lemma 6.14. ∎

###  6.3. $\gamma_{g}$-limits and colimits

In this paragraph we shall prove that any sequence
$({\mathcal{F}}_{n})_{n\in\mathbb{N}}$ which $\gamma_{g}$-converges to
${\mathcal{F}}$ can be turned into an inductive system whose homotopy colimit
is ${\mathcal{F}}$. We recall the notion of homotopy colimit in $D(N)$ (see
[BN93, Def. 2.1])

######  Definition 6.17.

Let $({\mathcal{F}}_{n},f_{n})_{n\in\mathbb{N}}$, with
$f_{n}\colon{\mathcal{F}}_{n}\xrightarrow[]{}{\mathcal{F}}_{n+1}$, be an
inductive system in $D(N)$; then $\operatorname{hocolim}_{n}{\mathcal{F}}_{n}$
is the cone of the morphism

$({\rm id}-f)\colon\bigoplus_{n\geq
0}{\mathcal{F}}_{n}\xrightarrow[]{}\bigoplus_{n\geq 0}{\mathcal{F}}_{n}$

where $f=\bigoplus_{n\geq 0}f_{n}$.

We thus have a distinguished triangle

(6.3) $\bigoplus_{n\geq 0}{\mathcal{F}}_{n}\xrightarrow[]{{\rm
id}-f}\bigoplus_{n\geq
0}{\mathcal{F}}_{n}\xrightarrow[]{p}\operatorname{hocolim}_{n}{\mathcal{F}}_{n}\xrightarrow[]{+1}$

However, like any cone, $\operatorname{hocolim}_{n}{\mathcal{F}}_{n}$ is only
defined up to (non unique) isomorphism and the morphism $p$ itself is not
uniquely defined. For a given triangle (6.3) and for any $n_{0}\in\mathbb{N}$,
we have a morphism
$i_{n_{0}}\colon{\mathcal{F}}_{n_{0}}\xrightarrow[]{}\operatorname{hocolim}_{n}{\mathcal{F}}_{n}$
defined as $i_{n_{0}}=p\circ j_{n_{0}}$ where $j_{n_{0}}$ is the obvious
morphism to the sum. For another choice of triangle, with $p^{\prime}$ instead
of $p$, there exists an automorphism $\psi$ of
$\operatorname{hocolim}_{n}{\mathcal{F}}_{n}$ such that $p^{\prime}=\psi\circ
p$. Then $i^{\prime}_{n_{0}}=p^{\prime}\circ j_{n_{0}}$ is “conjugate” to
$i_{n_{0}}$ in the sense $i^{\prime}_{n_{0}}=\psi\circ i_{n_{0}}$.

We recall the “nine diagram” from [BBD82, Prop. 1.1.11] (see [KS06, Ex. 10.6]
where it is stated precisely which arrows are given and which ones are
obtained), a variant of the octaedron axiom. We assume to be given a
commutative square as on the top left corner of the diagram below and four
distinguished triangles (two vertical and two horizontal) given by the
unmarked solid arrows. The arrows marked $[1]$ are just translations of
already defined solid arrows.
The proposition claims that the dotted arrows can be chosen to make all
squares commutative, except for the one with $Z^{2}$ at its top left corner,
which is anticommutative, and the new line and the new column are
distinguished triangles.

$\textstyle{X^{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{X^{0}[1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[1]}$$\textstyle{Y^{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y^{0}[1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[1]}$$\textstyle{Z^{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\ominus}$$\textstyle{Z^{0}[1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{-[1]}$$\textstyle{X^{0}[1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[1]}$$\textstyle{X^{1}[1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{[1]}$$\textstyle{X^{2}[1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{-[1]}$$\textstyle{X^{0}[2]}$

######  Remark 6.18.

Using the nine diagram built on the square

$\textstyle{\bigoplus_{n\geq
n_{0}}{\mathcal{F}}_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
id}-f}$$\textstyle{\bigoplus_{n\geq
n_{0}}{\mathcal{F}}_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bigoplus_{n\geq
0}{\mathcal{F}}_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\rm
id}-f}$$\textstyle{\bigoplus_{n\geq 0}{\mathcal{F}}_{n}}$

we can check that $\operatorname{hocolim}_{n\geq
0}{\mathcal{F}}_{n}\xrightarrow[]{{\raisebox{-2.58334pt}[0.0pt][-2.58334pt]{$\mspace{1.0mu}\sim\mspace{2.0mu}$}}}\operatorname{hocolim}_{n\geq
n_{0}}{\mathcal{F}}_{n}$. With the above notations, we take
$X^{0}=X^{1}=\bigoplus_{n\geq
n_{0}}{\mathcal{F}}_{n},Y^{0}=Y^{1}=\bigoplus_{n\geq
0}{\mathcal{F}}_{n},Z^{0}=Z^{1}=\bigoplus_{0\leq n\leq
n_{0}-1}{\mathcal{F}}_{n}$. The map from $Z^{0}$ to $Z^{1}$ is also equal to
${\rm id}-f$, but as is an upper triangular finite-dimensional matrix with
${\rm Id}$ on the diagonal, so it is invertible. This implies $Z^{2}=0$. As a
result the vertical arrow from $X^{2}$ to $Y^{2}$ is an isomorphism (in
$D(N)$).

A more general statement of the following Lemma can be found in Lemma 13.33.4
of [Stacks, Section 0A5K]. It tells us, as we might expect, that the limit of
a subsequence coincides with the limit of the original sequence.

######  Lemma 6.19.

Let $({\mathcal{F}}_{n},f_{n})_{n\in\mathbb{N}}$ be an inductive system in
$D(N)$ and $\sigma\colon\mathbb{N}\xrightarrow[]{}\mathbb{N}$ an increasing
map. Then the compositions of the $f_{n}$’s define an inductive system
$({\mathcal{F}}_{\sigma(n)},f^{\prime}_{n})$ and there exists an isomorphism
$j\colon\operatorname{hocolim}_{n}{\mathcal{F}}_{\sigma(n)}\xrightarrow[]{{\raisebox{-2.58334pt}[0.0pt][-2.58334pt]{$\mspace{1.0mu}\sim\mspace{2.0mu}$}}}\operatorname{hocolim}_{n}{\mathcal{F}}_{n}$
such that $j\circ i^{\prime}_{\sigma(n)}=i_{\sigma(n)}$, where
$i^{\prime}_{\sigma(n)}$ is the morphism $i_{\bullet}$ for
$({\mathcal{F}}_{\sigma(n)})$.

A family of morphisms
$u_{n}\colon{\mathcal{F}}_{n}\xrightarrow[]{}{\mathcal{G}}$ such that
$u_{n+1}\circ f_{n}=u_{n}$ induces a morphism
$u\colon\operatorname{hocolim}_{n}{\mathcal{F}}_{n}\xrightarrow[]{}{\mathcal{G}}$
such that $u\circ i_{n}=u_{n}$ for all $n$. Indeed we have
$\bigoplus_{n}u_{n}\circ({\rm id}-f)=0$, hence $\bigoplus_{n}u_{n}$ factorizes
as $\bigoplus_{n}u_{n}=u\circ p$. Again this morphism $u$ is not uniquely
defined.

######  Lemma 6.20.

Let $({\mathcal{F}}_{n},f_{n})_{n\in\mathbb{N}}$, with
$f_{n}\colon{\mathcal{F}}_{n}\xrightarrow[]{}{\mathcal{F}}_{n+1}$, be an
inductive system in $D(N)$. We let ${\mathcal{C}}(f_{n})$ be the cone of
$f_{n}$ and we assume that $\sum_{n}\gamma_{g}({\mathcal{C}}(f_{n}),0)$ is
finite. Then the sequence $({\mathcal{F}}_{n})_{n\in\mathbb{N}}$
$\gamma_{g}$-converges to $\operatorname{hocolim}_{k}{\mathcal{F}}_{k}$. More
precisely, if we let
$i_{n}\colon{\mathcal{F}}_{n}\xrightarrow[]{}\operatorname{hocolim}_{k}{\mathcal{F}}_{k}$
be the morphism defined after (6.3), we have
$\gamma_{g}({\mathcal{C}}(i_{n}),0)\leq
2\sum_{k=n}^{\infty}\gamma_{g}({\mathcal{C}}(f_{k}),0)$.

###### Proof.

Let $n_{0}$ be given. We set for short ${\mathcal{S}}=\bigoplus_{n\geq
n_{0}}{\mathcal{F}}_{n}$,
${\mathcal{C}}=\operatorname{hocolim}_{n}{\mathcal{F}}_{n}$. We checked that
${\mathcal{C}}$ is the cone of $u={\rm id}-\bigoplus_{n\geq
n_{0}}f_{n}\colon{\mathcal{S}}\xrightarrow[]{}{\mathcal{S}}$. We also consider
the similar sum but associated to the constant sequence
${\mathcal{G}}_{n}={\mathcal{F}}_{n_{0}}$ for all $n\geq n_{0}$ and
$\mathcal{T}=\bigoplus_{n\geq n_{0}}{\mathcal{G}}_{n}$. We define
$w\colon\mathcal{T}\xrightarrow[]{}\mathcal{T}$ as ${\rm id}-s$ where
$s_{n}:{\mathcal{G}}_{n}\longrightarrow{\mathcal{G}}_{n+1}$ is induced by
${\rm Id}_{{\mathcal{F}}_{n_{0}}}$. The cone of $w$ is
${\mathcal{F}}_{n_{0}}$. We let
$f_{n}^{m}\colon{\mathcal{F}}_{n}\xrightarrow[]{}{\mathcal{F}}_{m}$ be the
composition of the $f_{k}$’s and we define $F=\bigoplus_{n\geq
n_{0}}f_{n_{0}}^{n}\colon{\mathcal{T}}\xrightarrow[]{}{\mathcal{S}}$. Then
$F\circ w=u\circ F$ and the nine diagram recalled above gives the commutative
(except for one anticommuting square) diagram of distinguished triangles

$\textstyle{\mathcal{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{w}$$\scriptstyle{F}$$\textstyle{\mathcal{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q}$$\scriptstyle{F}$$\textstyle{{\mathcal{F}}_{n_{0}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j_{n_{0}}}$$\scriptstyle{+1}$$\textstyle{{\mathcal{S}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u}$$\textstyle{{\mathcal{S}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{{\mathcal{C}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{+1}$$\textstyle{{\mathcal{S}}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{+1}$$\textstyle{{\mathcal{S}}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{+1}$$\textstyle{{\mathcal{C}}(j_{n_{0}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{+1}$$\scriptstyle{+1}$

where ${\mathcal{S}}^{\prime}=\bigoplus_{n\geq
n_{0}}{\mathcal{C}}(f_{n_{0}}^{n})$, $j_{n_{0}}$ is conjugate to $i_{n_{0}}$
(unfortunately we cannot say they are equal because of the ambiguity in the
definition of the cone) and $\mathcal{C}(j_{n_{0}})$ is the cone of
$j_{n_{0}}$.

We set $r\colon{\mathcal{F}}_{n_{0}}\xrightarrow[]{}\mathcal{T}$ be the
inclusion of the first factor. The morphisms $r$ and $w$ give a splitting
$\mathcal{T}\simeq{\mathcal{F}}_{n_{0}}\oplus\mathcal{T}$ and it follows that
$q\circ r$ is an isomorphism. Since $i_{n_{0}}$ is “conjugate” to $p\circ
F\circ r$ (see after (6.3)), it is also conjugate to $j_{n_{0}}$ and
${\mathcal{C}}(j_{n_{0}})\simeq{\mathcal{C}}(i_{n_{0}})$.

Now we claim that
$\gamma_{g}({\mathcal{C}}(f_{n}^{m}),0)\leq\sum_{k=n}^{m-1}\gamma_{g}({\mathcal{C}}(f_{k}),0)$.
We prove this by induction on $m-n$. When $m=n+1$ this is clear. Since
$f_{n}^{m+1}=f_{m}\circ f_{n}^{m}$, we have, using Lemma 6.4 ( 6),
$\gamma_{g}({\mathcal{C}}(f_{n}^{m+1}),0)\leq\gamma_{g}({\mathcal{C}}(f_{n}^{m}),0)+\gamma_{g}({\mathcal{C}}(f_{m}),0)$.

In particular
$\gamma_{g}({\mathcal{C}}(f_{n_{0}}^{n}),0)\leq\varepsilon_{n_{0}}:=\sum_{k=n_{0}}^{\infty}\gamma_{g}({\mathcal{C}}(f_{k}),0)$,
for all $n\geq n_{0}$, and then
$\gamma_{g}({\mathcal{S}}^{\prime},0)\leq\varepsilon_{n_{0}}$. By the triangle
in the last row of the above diagram and Lemma 6.4 ( 5) we deduce
$\gamma_{g}({\mathcal{C}}(j_{n_{0}}),0)\leq 2\varepsilon_{n_{0}}$, as
required. ∎

######  Proposition 6.21.

Any Cauchy sequence in $D(N)$ with respect to $\gamma_{g}$ is convergent.

######  Remark 6.22.

The reader should be careful : the distance $\gamma_{g}$ is degenerate, so the
limit is by no means unique. However we shall see that the restriction of
$\gamma_{g}$ to the set of limits of constructible sheaves (i.e. in
$D_{lc}(N)$) the distance is non-degenerate (see Appendix B).

###### Proof.

Let $({\mathcal{F}}_{n})_{n\in\mathbb{N}}$ be a Cauchy sequence. Up to taking
a subsequence we can assume that
$\gamma_{g}({\mathcal{F}}_{n},{\mathcal{F}}_{m})\leq 2^{-n}$ for any $n\leq
m$. In particular, by Lemma 6.8, there exist
$f_{n}\colon{\mathcal{F}}_{n}\xrightarrow[]{}K_{\varphi}^{2^{-n}}({\mathcal{F}}_{n+1})$
such that $\gamma_{g}({\mathcal{C}}(f_{n}),0)\leq 2^{-n+1}$. We then set
$\varepsilon_{n}=\sum_{k\geq n}2^{-k}$ and
${\mathcal{F}}^{\prime}_{n}=K_{\varphi}^{-\varepsilon_{n}}({\mathcal{F}}_{n})$
so $f_{n}$ induces a map
$f^{\prime}_{n}:{\mathcal{F}}^{\prime}_{n}\longrightarrow{\mathcal{F}}^{\prime}_{n+1}$
and according to Lemma 6.20, ${\mathcal{F}}^{\prime}_{n}$ $\gamma_{g}$
converges to $\operatorname{hocolim}{\mathcal{F}}^{\prime}_{n}$. Since
$\gamma_{g}({\mathcal{F}}_{n},{\mathcal{F}}^{\prime}_{n})\leq\varepsilon_{n}$
we have that $({\mathcal{F}}_{n})_{n\geq 1}$ has the same limit. ∎

We just proved that to a Cauchy sequence we can associate a limit that is a
homotopy colimit. In the next Proposition we prove that conversely any
$\gamma_{g}$-limit of a sequence $({\mathcal{F}}_{n})_{n\in\mathbb{N}}$ is a
homotopy colimit of some (slightly modified) subsequence. We first notice an
easy consequence of Lemma 6.20.

######  Lemma 6.23.

Let ${\mathcal{F}}\in D(N)$ and let $\varepsilon_{n}>0$ be a sequence
decreasing to $0$. We have an inductive system
$(K_{\varphi}^{-\varepsilon_{n}}({\mathcal{F}}_{n}),\tau_{-\varepsilon_{n},-\varepsilon_{n+1}}({\mathcal{F}}))$.
Then the maps $\tau_{-\varepsilon_{n},0}({\mathcal{F}})$ induce an isomorphism
$\operatorname{hocolim}K_{\varphi}^{-\varepsilon_{n}}({\mathcal{F}}_{n})\xrightarrow[]{{\raisebox{-2.58334pt}[0.0pt][-2.58334pt]{$\mspace{1.0mu}\sim\mspace{2.0mu}$}}}{\mathcal{F}}$.

###### Proof.

Let
$\tau\colon\operatorname{hocolim}K_{\varphi}^{-\varepsilon_{n}}({\mathcal{F}}_{n})\xrightarrow[]{}{\mathcal{F}}$
be the morphism induced by the maps
$\tau_{-\varepsilon_{n},0}({\mathcal{F}})$. With the notation $i_{n}$ of Lemma
6.20 we have $\tau\circ i_{n}=\tau_{-\varepsilon_{n},0}({\mathcal{F}})$. By
the octaedron axiom there exists a distinguished triangle relating the cones
of these maps and Lemma 6.4 implies that
$\gamma_{g}(0,{\mathcal{C}}(\tau))\leq\gamma_{g}(0,{\mathcal{C}}(i_{n}))+\gamma_{g}(0,{\mathcal{C}}(\tau_{-\varepsilon_{n},0}({\mathcal{F}})))$.
By Lemma 6.8
$\gamma_{g}(0,{\mathcal{C}}(\tau_{-\varepsilon_{n},-\varepsilon_{n+1}}({\mathcal{F}})))\leq
2(\varepsilon_{n}-\varepsilon_{n+1})$ and we can apply Lemma 6.20. Hence
$\gamma_{g}(0,{\mathcal{C}}(i_{n}))$ goes to $0$ and
$\gamma_{g}(0,{\mathcal{C}}(\tau))$ is as small as desired. So
${\mathcal{C}}(\tau)\simeq 0$ and $\tau$ is an isomorphism. ∎

######  Proposition 6.24.

Let $({\mathcal{F}}_{n})_{n\in\mathbb{N}}$ be a sequence in $D(N)$ which
$\gamma_{g}$-converges to ${\mathcal{F}}$. Then, up to taking a subsequence,
there exist a sequence of positive numbers
$(\varepsilon_{n})_{n\in\mathbb{N}}$ converging to $0$ and morphisms
$f_{n}\colon
K_{\varphi}^{-\varepsilon_{n}}({\mathcal{F}}_{n})\xrightarrow[]{}K_{\varphi}^{-\varepsilon_{n+1}}({\mathcal{F}}_{n+1})$,
$u_{n}\colon
K_{\varphi}^{-\varepsilon_{n}}({\mathcal{F}}_{n})\xrightarrow[]{}{\mathcal{F}}$
such that $u_{n+1}\circ f_{n}=u_{n}$, for all $n$, and the morphism
$\operatorname{hocolim}K_{\varphi}^{-\varepsilon_{n}}({\mathcal{F}}_{n})\xrightarrow[]{}{\mathcal{F}}$
induced by the $u_{n}$’s (see after (6.3)) is an isomorphism.

###### Proof.

We set $\alpha_{n}=2^{-n}$. Up to taking a subsequence we can assume that
$\gamma_{g}({\mathcal{F}}_{n},{\mathcal{F}})\leq\alpha_{n+1}$ for each $n$.
This implies the existence of $0\leq\beta_{n}\leq\alpha_{n+1}$ and morphisms
$v_{n}\colon{\mathcal{F}}\xrightarrow[]{}K_{\varphi}^{\beta_{n}}({\mathcal{F}}_{n})$,
$w_{n}\colon
K_{\varphi}^{\beta_{n}}({\mathcal{F}}_{\sigma(n)})\xrightarrow[]{}K_{\varphi}^{\alpha_{n+1}}({\mathcal{F}})$
such that $w_{n}\circ v_{n}=\tau_{0,\alpha_{n+1}}({\mathcal{F}})$. Translating
these morphisms by $K_{\varphi}^{\bullet}(-)$ and alternating the
${\mathcal{F}}_{n}$ and the $K_{\varphi}^{-\alpha_{n}}({\mathcal{F}})$ we
obtain an inductive system

$\cdots\xrightarrow[]{}K_{\varphi}^{-\alpha_{n}}({\mathcal{F}})\xrightarrow[]{K_{\varphi}^{-\alpha_{n}}(v_{n})}K_{\varphi}^{-\varepsilon_{n}}({\mathcal{F}}_{n})\xrightarrow[]{K_{\varphi}^{-\alpha_{n}}(w_{n})}K_{\varphi}^{-\alpha_{n+1}}({\mathcal{F}})\\\
\xrightarrow[]{K_{\varphi}^{-\alpha_{n+1}}(v_{n+1})}K_{\varphi}^{-\varepsilon_{n+1}}({\mathcal{F}}_{n+1})\xrightarrow[]{K_{\varphi}^{-\alpha_{n+1}}(w_{n+1})}K_{\varphi}^{-\alpha_{n+2}}({\mathcal{F}})\xrightarrow[]{}\cdots$

where $\varepsilon_{n}=\alpha_{n}-\beta_{n}$. We define
$f_{n}=K_{\varphi}^{-\alpha_{n+1}}(v_{n+1})\circ
K_{\varphi}^{-\alpha_{n}}(w_{n})$ and
$u_{n}=\tau_{-\alpha_{n+1},0}({\mathcal{F}})\circ
K_{\varphi}^{-\alpha_{n}}(w_{n})$. The result follows from Lemmas 6.19 and
6.23. ∎

##  7\. Proof of Theorem 1.2

Let ${\mathcal{F}}\in D(N)$. We want to prove that around a point $z_{0}\in
SS({\mathcal{F}})$ there is no sequence $(\varphi_{j})_{j\geq 1}$ in
$\mathfrak{DHam}_{c}(B(z_{0},\varepsilon))$ such that
$\gamma-\lim_{j}\varphi_{j}={\rm Id}$ and $\varphi_{j}(SS({\mathcal{F}}))\cap
B(z_{0},\eta)=\emptyset$. We argue by contradiction and assume the existence
of such a sequence. Let ${\mathcal{K}}_{\varphi_{j}}$ be the sheaf in
$D^{b}(N\times N\times{\mathbb{R}})$ inducing $\varphi_{j}$ as recalled in §4.
We denote by $T^{*}_{\tau>0}(N\times\mathbb{R})$ the subset of
$T^{*}(N\times\mathbb{R})$ given by $\tau>0$ and we set $\rho\colon
T^{*}_{\tau>0}(N\times\mathbb{R})\xrightarrow[]{}T^{*}N$,
$(x,t,\xi,\tau)\mapsto(x,\xi/\tau)$. Then
$\rho(SS({\mathcal{K}}^{\varoast}_{\varphi_{j}}({\mathcal{F}}))\cap
T^{*}_{\tau>0}(N\times\mathbb{R}))=\varphi_{j}(SS({\mathcal{F}}))$ does not
intersect $B(z_{0},\eta)$. We want to prove that this implies that
$SS({\mathcal{F}})\cap B(z_{0},\eta)=\emptyset$.

We will use the following propositions.

######  Proposition 7.1.

Let $(\varphi_{j})_{j\geq 1}$ be a sequence in $\mathfrak{DHam}_{c}(T^{*}N)$
such that $\gamma-\lim\varphi_{j}={\rm Id}$ and
${\mathcal{K}}^{\varoast}_{\varphi_{j}}$ its quantization in $D^{b}(N\times
N\times{\mathbb{R}})$. Then for all ${\mathcal{F}}\in D^{b}(N)$ we have

$\gamma_{g}-\lim{\mathcal{K}}^{\varoast}_{\varphi_{j}}({\mathcal{F}})={\mathcal{K}}^{\varoast}_{{\rm
Id}}({\mathcal{F}})={\mathcal{F}}\boxtimes k_{[0,+\infty[}\in
D^{b}(N\times{\mathbb{R}})$

###### Proof.

This follows from Lemma 6.16 and Proposition 6.13. ∎

######  Remark 7.2.

Note that $\varphi_{j}(SS({\mathcal{F}}))$ is not a homogeneous manifold, and
$SS({\mathcal{K}}^{\varoast}_{\varphi_{j}}({\mathcal{F}}))\subset
T^{*}(N\times{\mathbb{R}})$ represents the homogenization of
$\varphi_{j}(SS({\mathcal{F}}))$.

Now for a sequence $(X_{j})_{j\geq 0}$ of subsets of a topological space $Z$,
its topological upper and lower limits are defined as

$\displaystyle\limsup_{j}X_{j}$
$\displaystyle=\bigcap_{n}\overline{\bigcup_{j\geq n}X_{j}}$
$\displaystyle=\left\\{x\in Z\mid\exists(x_{j})_{j\geq 1},\;x_{j}\in
X_{j}\;\text{for infinitely many}\;j,\;\lim_{j}x_{j}=x\right\\}$
$\displaystyle\liminf_{j}X_{j}$ $\displaystyle=\left\\{x\in
Z\mid\exists(x_{j})_{j\geq 1},x_{j}\in X_{j},\lim_{j}x_{j}=x\right\\}$

Obviously $\liminf_{j}X_{j}\subset\limsup_{j}X_{j}$. We have

######  Proposition 7.3.

Let $M$ be a manifold and let $g\colon T^{*}M\setminus
0_{M}\xrightarrow[]{}{\mathbb{R}}$ be a $1$-homogeneous autonomous Hamiltonian
function. If $({\mathcal{F}}_{j})_{j\geq 1}$ is a sequence such that
$\gamma_{g}-\lim{\mathcal{F}}_{j}={\mathcal{F}}$ then we have

$SS({\mathcal{F}})\subset\liminf_{j}SS({\mathcal{F}}_{j})$

###### Proof.

By Proposition 6.24, up to taking a subsequence and replacing
${\mathcal{F}}_{j}$ by $K_{\varphi_{g}}^{\varepsilon_{j}}(F_{j})$, for some
sequence $(\varepsilon_{j})$ converging to $0$, we can assume that there exist
morphisms $f_{j}\colon{\mathcal{F}}_{j}\xrightarrow[]{}{\mathcal{F}}_{j+1}$
such that
$\operatorname{hocolim}{\mathcal{F}}_{j}\xrightarrow[]{{\raisebox{-2.58334pt}[0.0pt][-2.58334pt]{$\mspace{1.0mu}\sim\mspace{2.0mu}$}}}{\mathcal{F}}$.
The homotopy colimit is defined by a distinguished triangle where the other
two terms are $\bigoplus_{n\geq n_{0}}{\mathcal{F}}_{n}$ (we can indeed
restrict to $\bigoplus_{n\geq n_{0}}$ by Remark 6.18). By the triangle
inequality for singular supports and Proposition 3.4, this implies that
$SS({\mathcal{F}})\subset\limsup_{j}SS({\mathcal{F}}_{j})$. Now let
$p=(x,\xi)\not\in\liminf_{j}SS({\mathcal{F}}_{j})$, we can find a subsequence
$\sigma(j)$, such that $p\notin\limsup_{j}SS({\mathcal{F}}_{\sigma(j)})$; but
${\mathcal{F}}_{\sigma(j)}$ has limit ${\mathcal{F}}$, so $p\notin
SS({\mathcal{F}})$. As a result we have
$SS({\mathcal{F}})\subset\liminf_{j}SS({\mathcal{F}}_{j})$. ∎

We may now conclude the proof of the Theorem.

As indicated above, we argue by contradiction and assume $SS({\mathcal{F}})$
is not $\gamma$-coisotropic at $z_{0}=(x_{0},p_{0})\in SS({\mathcal{F}})$.
Then there is a sequence $(\varphi_{j})_{j\geq 1}$ of Hamiltonian maps and an
open ball $B(z_{0},\eta)$ such that

  1. (1)

$(\varphi_{j})_{j\geq 1}$ $\gamma$-converges to ${\rm id}$,

  2. (2)

$\varphi_{j}(SS({\mathcal{F}}))\cap B(z_{0},\eta)=\emptyset$.

The first statement together with Proposition 7.1 implies that

${\mathcal{K}}^{\varoast}_{\varphi_{j}}({\mathcal{F}})\overset{\gamma_{g}}{\longrightarrow}{\mathcal{K}}^{\varoast}_{{\rm
id}}({\mathcal{F}})={\mathcal{F}}\boxtimes k_{[0,+\infty[}$

and Proposition 7.3, applied with $M=N\times\mathbb{R}$ and $g=\tau$, then
gives

$SS({\mathcal{F}})\times(\\{0\\}\times{\mathbb{R}})=SS({\mathcal{K}}^{\varoast}_{\rm
id}({\mathcal{F}}))\subset\liminf_{j}SS({\mathcal{K}}^{\varoast}_{\varphi_{j}}({\mathcal{F}}))$

Using $\rho(SS({\mathcal{K}}^{\varoast}_{\varphi_{j}}({\mathcal{F}}))\cap
T^{*}_{\tau>0}(N\times\mathbb{R}))=\varphi_{j}(SS({\mathcal{F}}))$ we deduce
$SS({\mathcal{F}})\subset\liminf_{j}\varphi_{j}(SS({\mathcal{F}}))$. In
particular $SS({\mathcal{F}})\cap B(z_{0},\eta)=\emptyset$ by the hypothesis
(2), which gives a contradiction.

##  8\.  $\gamma$-coisotropic vs cone-coisotropic

###  8.1. A $\gamma$-coisotropic set is cone-coisotropic

Our goal in this section is to prove Proposition 1.3.

Following Bouligand (see [Bou32]), one may define for a point $x$ in a subset
$V$ in a smooth manifold, two cones :

######  Definition 8.1.

The paratingent cone of a set $V$ at $x$ is

$C^{+}(x,V)=\left\\{\lim_{n}c_{n}(x_{n}-y_{n})\mid x_{n},y_{n}\in
V,c_{n}\in{\mathbb{R}}\;\lim_{n}x_{n}=\lim_{n}y_{n}=x,\lim_{n}c_{n}=+\infty\right\\}$

The contingent cone of a set $V$ at $x$ is

$C^{-}(x,V)=\left\\{\lim_{n}c_{n}(x_{n}-x)\mid x_{n}\in
V,c_{n}\in{\mathbb{R}},\;\lim_{n}x_{n}=x,\lim_{n}c_{n}=+\infty\right\\}$

Clearly $C^{-}(x,V)\subset C^{+}(x,V)$. Note that $C^{+}(x,V)$ is invariant by
$v\mapsto-v$, while it is not necessarily the case for $C^{-}(x,V)$. We then
have the following definition, which appears under the name “involutivity” in
the book of Kashiwara and Schapira

######  Definition 8.2 (Cone-coisotropic, see [KS90], definition 6.5.1 p.
271).

We shall say that $V$ is cone-coisotropic if whenever a hyperplane $H$ is such
that $C^{+}(x,V)\subset H$ then the symplectic orthogonal of $H$, $H^{\omega}$
is contained in $C^{-}(x,V)$.

\begin{overpic}[width=170.71652pt]{Coisotropic2-b.pdf}
\put(75.0,35.0){${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}C}$}
\put(12.0,45.0){${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}C}$}
\put(10.0,35.0){${H^{\omega}}$}
\end{overpic}\begin{overpic}[width=170.71652pt]{Coisotropic-b.pdf}
\put(75.0,35.0){${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}C}$}
\put(12.0,35.0){${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}C}$}
\put(27.0,10.0){${H^{\omega}}$} \end{overpic}

Figure 2. Coisotropic cone on the left, non-coisotropic on the right

###### Proof of Proposition 1.3.

We want to prove that if $V$ is not cone-coisotropic at $z\in V$ then it is
not $\gamma$-coisotropic at $z$. For $H$ a hyperplane we set $n_{H}$ the
normal to $H$ (for the usual scalar product). Let us assume that $z\in V$ is
such that $C^{+}(z,V)\subset H$ and $H^{\omega}\notin C^{-}(z,V)$. This will
follow from the next two lemmata. ∎

######  Lemma 8.3.

Let $V$ be a closed set, $z\in V$ and $H$ a hyperplane such that
$C^{+}(z,V)\subset H$. Then there exists a continuous function $f_{H,V}$
defined in a neighbourhood of $z$ in $H$ such that $f_{H,V}(z)=0$ and
$d_{G}f_{H,V}(z)=0$, where $d_{G}$ is the Gâteaux derivative, and such that
for any neighbourhood $C_{\varepsilon}^{+}(z,V)$ of $C^{+}(z,V)$ we have for
$\delta$ small enough

$V\cap B(z,\delta)\subset\left\\{z+x+f_{V}(x)n_{H}\mid x\in
C_{\varepsilon}^{+}(z,V)\right\\}$

###### Proof.

We can assume for simplicity that $z=0$. For $x\in H$ let
$V_{x}=V\cap(x+{\mathbb{R}}n_{H})$. We claim that for $x$ close enough to $0$,
$V_{x}$ contains at most one point. Indeed, if this was not the case, we would
have sequences $(y_{n})_{n\geq 1},(w_{n})_{n\geq 1}$ converging to $0$ such
that $y_{n}-w_{n}=t_{n}n_{H}$ with $t_{n}\neq 0$. But then by definition of
$C^{+}(0,V)$, we would have $n_{H}\in C^{+}(0,V)$ a contradiction.

As a result, on the closed set $\Gamma\subset H$ defined as the set of $x$
such that $V_{x}$ is non empty, we have a function $f$ such that

$x\in\Gamma\Longleftrightarrow(x,f(x))\in V$

Let us prove $f$ is continuous on $\Gamma$, at least in a neighbourhood of
$0$. If this was not the case, there would be a sequence $(z^{n})_{n\geq 1}$
converging to $0$ and for each $n$ two sequences $(x_{k}^{n})_{k\geq
1},\,(y_{k}^{n})_{k\geq 1}$ such that $\lim_{k}x_{k}^{n}=\lim y_{k}^{n}=z^{n}$
and $\lim_{k}f(x_{k}^{n})\neq\lim_{k}f(y_{k}^{n})$. Setting
$u_{k}^{n}=(x_{k}^{n},f(x_{k}^{n})),v_{k}^{n}=(y_{k}^{n},f(y_{k}^{n}))$ we
have

$u_{k}^{n}-v_{k}^{n}=(x_{k}^{n}-y_{k}^{n},f(x_{k}^{n})-f(y_{k}^{n}))$

which can be normalized so that, for any $n$, it has a limit $(0,1)$ when
$k\xrightarrow[]{}+\infty$. As a result for $k=k(n)$ large enough, setting
$u_{n}=u_{k(n)}^{n},v_{n}=v_{k(n)}^{n}$, there is a $\tau_{n}$ so that

$\|\tau_{n}(u_{n}-v_{n})-(0,1)\|<1/3$

so we can find a converging subsequence to a vector $w$ which does not belong
to $H$. Since $w\in C^{+}(0,V)$ we have a contradiction. From now on we
replace $\Gamma$ by $\Gamma\cap B(0,r)$ with $r$ small enough so that $f$ is
now continuous on $\Gamma$.

Let us now prove that
$\displaystyle\lim_{\begin{subarray}{c}|x|\xrightarrow[]{}0\\\
x\in\Gamma\end{subarray}}\frac{f(x)}{|x|}=0$. Indeed, if we had a sequence
$x_{k}$ in $\Gamma$ converging to $0$ and such that
$\displaystyle\lim_{k}\frac{f(x_{k})}{|x_{k}|}\neq 0$, the sequence of vectors
$\frac{1}{|x_{k}|}(x_{k},f(x_{k}))$ would have a limit which does not belong
to $H$, so $C^{+}(0,V)\not\subset H$, a contradiction.

We now will extend $f$ to $f_{H,V}$ such that

  1. (1)

$f_{H,V}(x)=f(x)$ for $x\in\Gamma$

  2. (2)

$f_{H,V}$ is continuous

  3. (3)

$d_{G}(f_{H,V}(0))=0$

For this we can assume for simplicity that $r=1$ and write $x=\rho\cdot\theta$
with $0\leq\rho\leq 1,\theta\in S^{n-1}$. We write
$g(s,\theta)=f(e^{s}\cdot\theta)$ with $s\in\mathopen{]}-\infty,0]$ and
identify $\Gamma$ with its image in $\mathopen{]}-\infty,0]\times S^{n-1}$.
Now let $g_{n}(s,\theta)$ be a continuous function defined on
$C_{n}=\mathopen{]}-n-1,-n+1\mathclose{[}\times S^{n-1}$ for $n\in\mathbb{N}$,
coinciding with $g$ on $\Gamma\cap C_{n}$ and having the same bound, i.e.

$\sup\left\\{g_{n}(s,\theta)\mid(s,\theta)\in
C_{n}\right\\}=\sup\left\\{g(s,\theta)\mid(s,\theta)\in\Gamma\cap
C_{n}\right\\}$ $\inf\left\\{g_{n}(s,\theta)\mid(s,\theta)\in
C_{n}\right\\}=\inf\left\\{g(s,\theta)\mid(s,\theta)\in\Gamma\cap
C_{n}\right\\}$

Such a function exists by Tietze’s extension theorem.

Now let $\chi_{n}$ be a partition of unity subordinated to the covering of
$\mathopen{]}-\infty,0]\times S^{n-1}$ by the $C_{n}$. We set

$G(s,\theta)=\sum_{n\geq 0}\chi_{n}(s)g_{n}(s,\theta)$

Note that this is well defined since whenever $g_{n}(s,\theta)$ is not
defined, we have $\chi_{n}(s,\theta)=0$, so we set
$\chi_{n}(s)g_{n}(s,\theta)=0$ outside of $C_{n}$. And since whenever
$g_{n}(s,\theta)$ is defined and $(s,\theta)\in\Gamma$ we have
$g_{n}(s,\theta)=g(s,\theta)$, we see that $F$ is an extension of $G$. We now
set

$f_{H,V}(e^{s}\cdot\theta)=G(s,\theta)$

Properties (1) and (2) are then obvious. As for the third one, we notice that

$\sup\left\\{f_{H,V}(x)\mid|x|\leq
e^{-n}\right\\}=\sup\left\\{G(s,\theta)\mid(s,\theta)\mid
s\leq-n\right\\}\leq\\\ \sup_{k\geq
n}\sup\left\\{g_{k}(s,\theta)\mid(s,\theta)\in
C_{k}\right\\}\leq\sup\left\\{g(s,\theta)\mid(s,\theta)\in\Gamma,\,s\leq-n\right\\}=\\\
\sup\left\\{f(x)\mid x\in\Gamma,\,|x|\leq e^{-n}\right\\}$

We see that since
$\displaystyle\lim_{\begin{subarray}{c}|x|\xrightarrow[]{}0\\\
x\in\Gamma\end{subarray}}\frac{f(x)}{|x|}=0$ we have
$\displaystyle\lim_{x\xrightarrow[]{}0}\frac{f_{H,V}(x)}{|x|}=0$.

As a result

$\lim_{t\xrightarrow[]{}0}\frac{1}{t}(f_{H,V}(th)-f_{H,V}(0))=0$

and we may conclude that $d_{G}f_{H,V}(z)=0$. ∎

######  Lemma 8.4.

Let $f$ be a continuous function on the hyperplane $H$ such that $d_{G}f(0)=0$
and $C$ a cone in $H$ not containing $H^{\omega}$. Set

$\widetilde{C}=\left\\{x+f(x)n_{H}\mid x\in C\right\\}$

Then for any positive $\varepsilon$, there is a Hamiltonian map $\varphi$ such
that $\gamma(\varphi)\leq\varepsilon$ and

$\varphi(\widetilde{C}\cap B(0,1))\cap B(0,1)=\emptyset$

###### Proof.

We shall assume w.l.o.g. that $H=\\{p_{n}=0\\}$ and $z=0$, so that
$H^{\omega}$ is given by the direction $q_{n}$. First let us mention that the
case $f=0$ is trivial. Just take a Hamiltonian $H(p_{n})$ such that $H(0)=0$,
$H^{\prime}(0)=A$ and $H$ is $C^{0}$ small, $A$ being chosen so that in
$B(0,1)$ we have

$C\cap B(0,1)\cap(A\frac{\partial}{\partial q_{n}}+C)=\emptyset$

\begin{overpic}[width=227.62204pt]{Pairofcones2-b.pdf} \end{overpic} Figure 3.
The cone $C$ and its translation by $A\frac{\partial}{\partial q_{n}}$

This extends to the case where $f$ is smooth. Indeed, let us set
$K(q_{n},f(q_{n},\bar{q},\bar{p}),\bar{q},\bar{p})=0$ and $\frac{\partial
K}{\partial p_{n}}(q_{n},f(q_{n},\bar{q},\bar{p}),\bar{q},\bar{p})=A$. We
have, for any $(q_{n},p_{n},\bar{q},\bar{p})\in\widetilde{C}$,
$\varphi_{H}^{t}(q_{n},p_{n},\bar{q},\bar{p})=(q_{n}+At,p_{n}(t),\bar{q}(t),\bar{p}(t))$
hence

$\varphi_{K}^{t}(\widetilde{C})\cap B(0,1)=\emptyset$

Of course we have to truncate the flow, so that it is unchanged on the
trajectory of points in $\varphi_{K}^{t}(\widetilde{C})\cap B(0,1)$, but this
is not difficult.

Note that the same holds if we replace $\widetilde{C}$ by the region

$\widetilde{C}_{g,g^{\prime}}=\left\\{(q_{n},p_{n},\bar{q},\bar{p})\mid(q_{n},\bar{q},\bar{p})\in
C,g(q_{n}\bar{q},\bar{p})\leq p_{n}\leq
g^{\prime}(q_{n}\bar{q},\bar{p})\right\\}$

where $g\leq g^{\prime}$ are smooth functions with $dg(0)=dg^{\prime}(0)=0$,
except that now we cannot have $\gamma(\varphi_{K})$ arbitrarily small because
$K$ has oscillation bounded from below by $A\|g-g^{\prime}\|_{C^{0}}$. Indeed
we must have $\frac{\partial K}{\partial p_{n}}=A$ on a segment in the $p_{n}$
direction of length
$g^{\prime}(q_{n}\bar{q},\bar{p})-g^{\prime}(q_{n}\bar{q},\bar{p})$. However
we can assume $\gamma(\varphi_{K})\leq A\|g-g^{\prime}\|_{C^{0}}+\varepsilon$
whith $\varepsilon>0$ arbitrarily small.

The general case is obtained as follows. We take a sequence $g_{k}\leq f\leq
g^{\prime}_{k}$ such that $\|g_{k}-g^{\prime}_{k}\|_{C^{0}(B(0,r))}$ goes to
zero with $k$. We can then displace $\widetilde{C}_{g_{k},g^{\prime}_{k}}$
hence $\widetilde{C}$ by a map with $\gamma(\varphi_{k})\leq
A\|g_{k}-g^{\prime}_{k}\|_{C^{0}(B(0,r))}+\varepsilon_{k}$. As $k$ goes to
infinity, we may let $\varepsilon_{k}$ go to zero. This concludes our proof. ∎

######  Remark 8.5.

The proof actually shows that $V$ is locally rigid in the sense of Usher (see
[Ush19]). So we have that $\gamma$-coisotropic implies locally rigid which
implies cone-coisotropic.

###  8.2. A cone-coisotropic set that is not $\gamma$-coisotropic

Let $K_{a}$ be the central Cantor set of ratio $a$. In other words we set

$\displaystyle F_{1}=[0,a]\cup[1-a,1]$ $\displaystyle
F_{2}=[0,a^{2}]\cup[a(1-a),a]\cup[1-a,1-a+a^{2}]\cup[1-a^{2},1]$

and $K_{a}=\bigcap_{n}F_{n}$.

######  Proposition 8.6.

For $a<2^{-2n}$, the set $X=K_{a}^{2n}$ is cone-coisotropic but not
$\gamma$-coisotropic.

###### Proof.

We claim that $C^{+}(z,X)$ is not contained in any hyperplane. Indeed for
$x\in K=K_{a}$ the cone $C^{+}(x,K)$ is never empty since any point in $K$ is
a cluster point of $K$. In fact $C^{-}(x,K)$ is either
${\mathbb{R}}_{-},{\mathbb{R}}_{+}$ or ${\mathbb{R}}$ depending whether $x$ is
the limit of points of $K$ less than $x$, the limit of points of $K$ greater
than $x$ or both, and $C^{+}(x,K)=\mathbb{R}$. Even though the paratingent
cone of a product does not necessarily contain the product of the paratingent
cones of the factors, in the present case $C^{+}(z,X)$ contains the rays
$\\{0^{j-1}\\}\times C^{+}(x_{j},K)\times\\{0^{n-j}\\}$ where
$z=(x_{1},..,x_{2n})$, and $0^{k}=(0,...,0)\in{\mathbb{R}}^{k}$. Therefore
$C^{+}(z,X)$ is not contained in any hyperplane and $X$ is trivially cone-
coisotropic.

We now claim $X$ is not $\gamma$-coisotropic. Pick $z\in X$ and balls
$B(z,\varepsilon)$, $B(z,\eta)$, $0<\eta<\varepsilon$. Let $k$ big enough (to
be made precise later). Since $K$ is contained in $F_{k}$ which consists of
$2^{k}$ intervals of length $a^{k}$, our $X$ is contained in a family
${\mathcal{I}}$ of cubes of edge length $a^{k}$ with
$|{\mathcal{I}}|=2^{2nk}$. The projection
$(q,p)\mapsto(q_{2},\dots,q_{n},p_{1},\dots,p_{n})$ maps ${\mathcal{I}}$ to a
family ${\mathcal{I}}^{\prime}$ of cubes of $\mathbb{R}^{2n-1}$ (again these
cubes have edge length $a^{k}$ and $|{\mathcal{I}}^{\prime}|=2^{(2n-1)k}$). We
choose disjoints neighbourhoods $U_{i}$, $i\in{\mathcal{I}}^{\prime}$, of the
cubes in ${\mathcal{I}}^{\prime}$. For a given
$i_{0}\in{\mathcal{I}}^{\prime}$, let $C_{j}$, $j=1,\ldots,N$, be the cubes of
${\mathcal{I}}$ contained in $B(z,\varepsilon)\cap(\mathbb{R}\times
U_{i_{0}})$, ordered according to the $q_{1}$ variable. Let $N_{1}$ be the
maximal index $j$ such that $C_{j}$ meets $B(z,\eta)$. We push all cubes
$C_{2},\dots,C_{N_{1}}$ to the left, close to $C_{1}$, in the space between
$B(z,\varepsilon)$ and $B(z,\eta)$ (if $k$ is big enough), as follows. We move
$C_{2}$ by a translation $T_{2}$ with direction $-\partial_{q_{1}}$, so that
$T_{2}(C_{2})$ is close to $C_{1}$. Then we move $C_{3}$ by a similar
translation $T_{3}$, so that $T_{3}(C_{3})$ is close to $T_{2}(C_{2})$. We go
on until $C_{N_{1}}$. We have thus accumulated at most $2^{k}$ cubes of edge
length $a^{k}$ near $C_{1}$. Since $a<1/2$, we have room to put all these
cubes in $B(z,\varepsilon)\setminus B(z,\eta)$ for $k$ big enough.

Now the translation $T_{j}$ can be realized by a Hamiltonian function of the
form $h_{j}=h(p_{1})\rho_{j}(q,p)$, where $h^{\prime}=1$ and $\rho_{j}$ is a
bump function vanishing outside $B(z,\varepsilon)\cap(\mathbb{R}\times
U_{i_{0}})$ and over $C_{1}\cup\dots\cup T_{j-1}(C_{j-1})\cup
C_{j+1}\cup\dots\cup C_{N_{1}}$ and equals to $1$ over the path swept by
$C_{j}$. We have $||h||\sim a^{k}$ and we apply the isotopy for a length of
time less than $\varepsilon<1$. We have to do this less than $|{\mathcal{I}}|$
times. Since we chose $a<2^{-2n}$, the composition of all these isotopies has
norm as small as required.

\begin{overpic}[width=170.71652pt]{Non-gammab2.pdf}
\end{overpic}\begin{overpic}[width=170.71652pt]{Non-gamma2b2.pdf}
\end{overpic}

Figure 4. Moving the cubes..

∎

######  Remark 8.7.

We can weaken the assumption $a<2^{-2n}$ to $a<1/4$ at the expense of a more
complicated proof.

######  Question 8.8.

Find an example where the image of a smooth non-coisotropic submanifold $C$ by
a symplectic homeomorphism, $\varphi$, is such that $\varphi(C)$ is cone-
coisotropic (but will not be $\gamma$-coisotropic by [Vit22]).

##  9\. Questions and comments

We gather in this section a number of questions that sprang up naturally while
writing this paper.

######  Question 9.1.

What is the $\gamma_{g}$-distance of $k_{X}$ and $k_{Y}$ for two closed sets
$X,Y$. If $X,Y$ do not have the same cohomology, the distance is infinite. So
we can focus on the case when $Y=X_{\varepsilon}$ is an
$\varepsilon$-neighbourhood of $X$. When is it true that
$\gamma_{g}(k_{X},k_{X_{\varepsilon}})\leq C\varepsilon$ ? Or when do we have
$\gamma-\lim k_{X_{\varepsilon}}=k_{X}$ ?

For the next four questions we choose on $N$ a real analytic structure. We let
$D_{lc}(N)$ be the category of limits of constructible sheaves considered in
Appendix B.

######  Question 9.2.

In Appendix B we describe the objects of $D_{lc}(\mathbb{R})$ with
microsupport contained in $\\{\tau\geq 0\\}$. Is there a similar description
in the general case ?

######  Question 9.3.

Using Lemma B.3 and the proof of Lemma B.1 it is probably not hard to see that
$D_{lc}(N)$ is a triangulated category. If we have a map $f\colon
N\xrightarrow[]{}M$, can we find conditions so that $Rf_{*}(D_{lc}(N))\subset
D_{lc}(M)$, $f^{-1}(D_{lc}(M))\subset D_{lc}(N)$?

For example this is true for a map of the type $f\times{\rm
id}_{\mathbb{R}}\colon N\times\mathbb{R}\xrightarrow[]{}M\times\mathbb{R}$, if
we consider the $\gamma_{\tau}$-distance, because $Rf_{*}$ and $f^{-1}$ are
Lipschitz for this distance. In general $f^{-1}$ is not Lipschitz: take
$f\colon\mathbb{R}\xrightarrow[]{}\mathbb{R}^{2}$, $x\mapsto(x,0)$, and
$D_{\varepsilon}$ the closed disc with center $(0,1)$ and radius
$1-\varepsilon$; then $\gamma_{g}(k_{D_{0}},k_{D_{\varepsilon}})=\varepsilon$
but $\gamma_{g}(f^{-1}(k_{D_{0}}),f^{-1}(k_{D_{\varepsilon}}))=+\infty$ (here
$g(q,p)=||p||$). Hence it is not clear that $f^{-1}(D_{lc}(M))\subset
D_{lc}(N)$.

######  Question 9.4.

Let $D_{dom}(N)$ be the subcategory of $D(N)$ generated in the triangulated
sense by the $k_{U}$, where $U$ runs over the domains with smooth boundary. It
should not be difficult to prove that $D_{dom}(N)\subset D_{lc}(N)$ by
approximating domains with smooth boundary by domains with analytic boundary.
Is it true that, for any subanalytic open subset $V$ of $X$ and any
$\varepsilon>0$, there exists a domain $U$ with smooth boundary such that
$U\subset V$ and $K_{\varphi_{g}}^{\varepsilon}(k_{U})\simeq
k_{U_{\varepsilon}}$ with $U_{\varepsilon}$ open and $V\subset
U_{\varepsilon}$ ? In this case we can see that $D_{lc}(N)$ coincides with the
$\gamma$-closure of $D_{dom}(N)$, which would imply that $D_{lc}(N)$ only
depends on the smooth structure of $N$.

######  Question 9.5.

Let $D_{tri}(N)$ be the set of sheaves which are constructible with respect to
some (non-fixed !) triangulation of $N$. Do the $\gamma$-closures of
$D_{dom}(N)$ and $D_{tri}(N)$ coincide ? Are they equal to $D_{lc}(N)$?

We remind the reader of the following (see [KS18], theorem 2.11, and [PS21],
corollary 2.4.2 )

######  Proposition 9.6.

We have for all $1$-Lipschitz maps $f:M\longrightarrow N$ the inequality

$\gamma_{g}((Rf)_{*}{\mathcal{F}},(Rf)_{*}{\mathcal{G}})\leq\gamma_{g}({\mathcal{F}},{\mathcal{G}})$

######  Question 9.7.

Can we give conditions on ${\mathcal{F}}$,${\mathcal{G}}$ (for example
$\gamma_{g}({\mathcal{F}},{\mathcal{G}})<\infty$) such that the opposite
inequality

$\gamma_{g}({\mathcal{F}},{\mathcal{G}})\leq C\sup_{f\in
Lip_{1}(X,{\mathbb{R}})}\gamma_{|\tau|}(Rf_{*}({\mathcal{F}}),Rf_{*}({\mathcal{G}}))$

holds for some constant $C$?

######  Question 9.8.

Let $f$ be a continuous function on $N$. Then
${\mathcal{F}}_{f}=k_{\\{(x,t)\mid f(x)\leq t\\}}$ is a sheaf on
$N\times{\mathbb{R}}$, and Vichery defined a subdifferential as

$\partial f=SS({\mathcal{F}}_{f})\cap\\{\tau=1\\}/(t)\subset T^{*}N$

Its intersection with $T_{x}^{*}N$ is $\partial f(x)$. On the other hand
$df\in\widehat{\mathfrak{L}}(T^{*}N)$, so we may ask whether

$SS({\mathcal{F}}_{f})\cap\\{\tau=1\\}/(t)=\partial f=\gamma-{\rm supp}(df)$

This leads to more general questions. First notice the following

######  Proposition 9.9.

There is a functor $Q:\widehat{\mathcal{L}}(T^{*}N)\longrightarrow
D_{lc}(N\times{\mathbb{R}})$ extending the usual quantization from [Gui12,
Vit19]. Moreover $Q$ is bi-Lipschitz of ratio $2$ from
$(\widehat{\mathcal{L}}(T^{*}N),\gamma)$ to
$(D_{lc}(N\times{\mathbb{R}}),\gamma_{\tau})$.

###### Proof.

Let $L_{n}$ be a Cauchy sequence in $\mathcal{L}(T^{*}N)$. Then
${\mathcal{F}}_{L_{n}}$ is a Cauchy sequence in $D_{lc}(N\times{\mathbb{R}})$.
The map $L\longrightarrow{\mathcal{F}}_{L}$ is a bi-Lipschitz map from the
metric $c$ to the (pseudo-) metric $\gamma_{\tau}$. As we saw in Remark 6.12,
the metric $\gamma_{\tau}$ is bounded between $1/2c$ and $c$ where $c$ is the
metric defined on $\mathcal{L}(T^{*}N)$ in [Vit22] by
$c(\widetilde{L}_{1},\widetilde{L}_{2})=|c_{+}(L_{1},L_{2})|+|c_{-}(L_{1},L_{2})|$.
Since according to Proposition B.7 the metric $\gamma_{\tau}$ is a bona fide
metric in $D_{lc}(N\times{\mathbb{R}})$, limits are unique and the limit of
${\mathcal{F}}_{L_{n}}$ is well-defined. ∎

######  Question 9.10.

Set $\rho(x,t;\xi,\tau)=(x;\xi/\tau)$. Given
$\Lambda\in\widehat{\mathcal{L}}(T^{*}N)$, do we have $\gamma-{\rm
supp}(\Lambda)=\rho(SS^{\bullet}(Q(\Lambda)))$ ?

A positive answer would also imply a positive answer to Question 9.8, i.e.
$\partial f=\gamma-{\rm supp}(df)$ by taking $\Lambda_{n}=df_{n}$ for some
sequence of differentiable functions converging to $f$.

######  Question 9.11.

Can one characterize the sheaves ${\mathcal{F}}$ such that there exists an
element $\widehat{SS}({\mathcal{F}})$ in $\widehat{\mathcal{L}}(T^{*}N)$ so
that $SS({\mathcal{F}})=\gamma-{\rm supp}(\widehat{SS}({\mathcal{F}}))$ ? In
the previous question we considered the ${\mathcal{F}}$ which are
$\gamma$-limits of sheaves ${\mathcal{F}}_{j}$ quantizing some exact
Lagrangian.

Let us give an example of such a situation : let $M,N$ be closed manifolds.
For a map $f\in C^{\infty}(M,N)$ set $\Lambda_{f}$ be the correspondence in
$T^{*}M\times T^{*}N$ defined by

$\Lambda_{f}=\left\\{(x,p_{x},y,p_{y})\mid y=f(x),p_{y}\circ
df(x)=-p_{x}\right\\}$

Now let $f_{n}$ be a sequence of smooth maps $C^{0}$-converging to $f\in
C^{0}(M,N)$. For ${\mathcal{F}}\in D^{b}(M)$ the sequence
$(R(f_{n})_{*}{\mathcal{F}})_{n\geq 1}$ $\gamma$-converges to
$Rf_{*}({\mathcal{F}})$ and
$SS(R(f_{n})_{*}{\mathcal{F}})\subset\Lambda_{f_{n}}\circ SS({\mathcal{F}})$.
Thus $SS(R(f_{n})_{*}{\mathcal{F}})$ ) should $\gamma$-converge to a subset of
$\Lambda_{f}\circ SS({\mathcal{F}})$. Of course since $f$ is only $C^{0}$,
$\Lambda_{f}$ is not defined as a set, but it is well defined in
$\widehat{\mathcal{L}}(T^{*}N\times T^{*}M)$, hence $\Lambda_{f}\circ
SS({\mathcal{F}})\in\widehat{\mathcal{L}}(T^{*}N)$. Thus we expect
$\widehat{SS}(Rf_{*}({\mathcal{F}}))=\Lambda_{f}\circ SS({\mathcal{F}})$ and
$SS(Rf_{*}{\mathcal{F}})\subset\gamma-{\rm supp}(\Lambda_{f}\circ
SS({\mathcal{F}}))$.

######  Question 9.12.

Let ${\mathcal{F}}_{n}$ be a sequence in $D(N)$ $\gamma_{g}$-converging to
${\mathcal{F}}_{\infty}$. We set $\Lambda_{n}=SS({\mathcal{F}}_{n})$ and we
assume that $\Lambda_{n}$ Hausdorff converges to $\Lambda_{\infty}$. We know
that $SS({\mathcal{F}}_{\infty})\subset\Lambda_{\infty}$ but when is it true
that $\Lambda_{\infty}=SS({\mathcal{F}}_{\infty})$? We can prove it if
$\Lambda_{\infty}$ is a smooth Lagrangian or more generally minimally
$\gamma$-coisotropic. Indeed Proposition 7.3 shows that
$SS({\mathcal{F}}_{\infty})\subset\Lambda_{\infty}$ and the involutivity of
the singular support gives the equality (by an argument similar to the one in
the proof of Proposition 9.13). Assuming $N=M\times\mathbb{R}$ and the
${\mathcal{F}}_{n}$ are quantizations of exact Lagrangian, Proposition 9.13
shows that this still hold when $\Lambda_{\infty}$ is the conification of some
$\varphi(L_{0})$ for $L_{0}$ smooth Lagrangian, $\varphi$ a symplectic
homeomorphism.

Note however that the answer cannot be positive in general: Let $f(x)$ be a
smooth bounded function on $\mathbb{R}$, such that $\max{f^{\prime}}=1$,
$\min{f^{\prime}}=-1$. Let $f_{n}(x)=f(nx)/n$ and
${\mathcal{F}}_{n}=k_{\\{t\geq f_{n}(x)\\}}$. Then ${\mathcal{F}}_{n}$
converges to ${\mathcal{F}}_{\infty}=k_{\mathbb{R}\times[0,\infty[}$ but
$SS({\mathcal{F}}_{n})$ converges to a set $\Lambda_{\infty}$ bigger than
$SS(F_{\infty})$. In fact $\Lambda$ is the cone over a “cross” $C$ in
$T^{*}\mathbb{R}$ with $C=0_{\mathbb{R}}\cup(\\{0\\}\times[-1,1])$.

######  Proposition 9.13.

Let $L=\varphi(L_{0})$ where $L_{0}$ is smooth Lagrangian and
$\varphi\in\mathcal{H}_{\gamma}(M,\omega)=\widehat{\mathfrak{DHam}}(M,\omega)\cap
Homeo(M)$. Then any closed proper subset of $L$ is not $\gamma$-coisotropic.

###### Proof.

Let $L^{\prime}\subset L$. Since being $\gamma$-coisotropic is invariant by
$\widehat{\mathfrak{DHam}}(M,\omega)\cap Homeo(M)$ it is enough to prove the
Proposition for $L_{0}$ i.e. in the smooth case. Now there is a ball $B(z,r)$
in $L$ such that $\overline{B(z,r)}\cap L^{\prime}=\\{z^{\prime}\\}$. Moreover
we may choose $r$ to be arbitrarily small. Let us consider a flow on $L$ such
that $\rho^{t}(B(z,r)\subset B(z^{\prime},r)$. In a chart containing $B(z,2r)$
we may assume that $\rho^{t}$ is a translation. Then $\rho^{1}(L^{\prime})\cap
B(z^{\prime},r)=\emptyset$. Since if $X_{t}$ is the vector field generating
$\rho^{t}$, the Hamiltonian written in action-angle coordinates (i.e. $L$ is
given locally by $p=0$) can be chosen of the form $\chi(|p|)\langle
p,X_{t}(x)\rangle$ for some bump function $\chi$, and this can be made
arbitrarily small. As a result, $L^{\prime}$ is not $\gamma$-coisotropic.

\begin{overpic}[width=170.71652pt]{Guill-2b.pdf}
\put(45.0,42.0){${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}z^{\prime}}$}
\put(15.0,45.0){${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\varphi(L)}$}
\put(45.0,25.0){$z$} \put(15.0,25.0){$L$} \end{overpic} Figure 5.  Proof of
Proposition 9.13

∎

######  Question 9.14.

Let $V$ be $\gamma$-coisotropic of dimension $n$. Then for $z\in V$ and
$\varepsilon$ small enough we have $V\setminus B(z,\varepsilon)$ is not
coisotropic.

##  Appendix A  Persistence modules and Barcodes

For this section we refer also to [Zha20]

We have already noticed after Definition 5.3 that the spectral invariants of
$L_{1},L_{2}\in\mathcal{L}(T^{*}N)$ are encoded in the sheaf over $\mathbb{R}$
given by
$\mathcal{H}=Rt_{*}\operatorname{\mathcal{H}om}^{\ast}({\mathcal{F}},{\mathcal{G}}))$,
where $t\colon N\times\mathbb{R}\xrightarrow[]{}\mathbb{R}$ is the projection,
because of the isomorphism

$\operatorname{R\Gamma}_{N\times\\{t\\}}(N\times\mathbb{R};\operatorname{\mathcal{H}om}^{\ast}({\mathcal{F}}_{L_{1}},{\mathcal{F}}_{L_{2}})))\simeq\operatorname{R\Gamma}_{\\{t\\}}(\mathbb{R};\mathcal{H})$

We recall that $\mathcal{H}$ belongs to the category $D_{\tau\geq
0}(\mathbb{R})$ of sheaves with singular support in $\\{\tau\geq 0\\}$.

We remark that $D_{\tau\geq 0}(\mathbb{R})$ contains the derived category of
persistence modules, where by a persistence module, we mean a constructible
sheaf on $({\mathbb{R}},\leq)$. We refer to [Bar94, Cha+09, ELZ02, KS18, ZC05]
for the theory and applications. Such a persistence module is uniquely defined
by the finite-dimensional vector spaces $V_{t}=\mathcal{F}(]-\infty,t[)$ and
the linear maps $r_{s,t}:V_{t}\longrightarrow V_{s}$ defined for $s\leq t$,
such that

  1. (1)

for $s<t<u$ we have $r_{u,t}\circ r_{s,t}=r_{s,u}$,

  2. (2)

$\lim_{t>s}V_{t}=V_{s}$ where the limit is that of the directed system given
by the $r_{s,t}$,

  3. (3)

$r_{t,t}={\rm Id}$.

It is a well-known result that persistence modules have a decomposition as sum
of barcodes : this is Gabriel’s theorem ([Gab72] on quivers in the finite case
and Crawley-Boevey’s theorem ([Cra15]) in the locally finite case. More
precisely, using Gabriel’s theorem and the fact that any complex of objects in
an Abelian category of homological dimension $1$ is the sum of its cohomology,
we find: any ${\mathcal{F}}\in D_{\tau\geq 0}(\mathbb{R})$ which is
constructible for a finite stratification can be written
${\mathcal{F}}\simeq\bigoplus_{j\in{\mathcal{I}}}k_{[a_{j},b_{j}[}[-n_{j}]$,
where ${\mathcal{I}}$ is a finite family, and this decomposition is unique up
to isomorphism. (The hypothesis that the singular support is in $\\{\tau\geq
0\\}$ is useless here.) We remark that the stratification consists only of a
finite set of points in $\mathbb{R}$ and the connected components of its
complement, where ${\mathcal{F}}$ is constant. This finite set is thus the
projection to $\mathbb{R}$ of $SS^{\bullet}({\mathcal{F}})$. If
$L\in\mathcal{L}(T^{*}N)$ is generic in the sense that $L$ meets $0_{N}$
transversally, then $Rt_{*}({\mathcal{F}}_{L})$ is of the above type, where
the stratification is given by the set of actions of the points in $L\cap
0_{N}$.

In [KS18] it is explained how to go from the case of a finite stratification
to the case of a locally finite one (then ${\mathcal{I}}$ is at most
countable). By [Cra15] the same result holds in fact (at least in $D_{\tau\geq
0}(\mathbb{R})$) under the more general hypothesis that ${\mathcal{F}}_{x}$ is
finite dimensional for all $x\in\mathbb{R}$. However the sheaves we encounter
in this paper are not necessarily of this type: if $f$ is the function
$f(x)=x^{2}\sin(1/x)$ and ${\mathcal{F}}=Rt_{*}(k_{\\{t\geq f(x)\\}})$, then
${\mathcal{F}}_{0}$ is infinite dimensional. But, by a small perturbation of
$L$, we see that all sheaves of the form $Rt_{*}({\mathcal{F}}_{L})$ are in
the closure of the set of constructible sheaves. Using Corollary B.11 we
deduce:

######  Proposition A.1.

Let $L_{1},L_{2}\in\mathcal{L}(T^{*}N)$. Then there exists an at most
countable family ${\mathcal{I}}$ (which is finite in the case of transverse
intersection) such that

$(Rt)_{*}(\operatorname{\mathcal{H}om}^{\ast}({\mathcal{F}}_{L_{1}},{\mathcal{F}}_{L_{2}}))=\bigoplus_{j\in{\mathcal{I}}}k_{[a_{j},b_{j}[}[-n_{j}],$

where for each $j\in{\mathcal{I}}$ we have
$a_{j}\in\mathbb{R}\cup\\{-\infty\\}$, $b_{j}\in\mathbb{R}$,
$n_{j}\in\mathbb{Z}$. Moreover there is a unique $j_{-}$ such that
$a_{j_{-}}=-\infty$ and $n_{j_{-}}=-1$ and then $b_{j_{-}}=c_{-}(L_{1},L_{2})$
and a unique $j_{+}$ such that $a_{j_{+}}=+\infty$ and $n_{j_{+}}=n-1$ and
then $b_{j_{+}}=c_{+}(L_{1},L_{2})$.

More generally, with the notations of the proposition we let
${\mathcal{I}}_{\infty}$ be the set of $j$ such that $a_{j}=-\infty$. Then
$H^{*}_{t_{0}}((Rt)_{*}(\operatorname{\mathcal{H}om}^{\ast}({\mathcal{F}}_{L_{1}},{\mathcal{F}}_{L_{2}})))\simeq\bigoplus_{j\in{\mathcal{I}}_{\infty}}k[-n_{j}]$,
for $t_{0}\ll 0$, and we deduce that the spectral invariants
$c(\alpha,L_{1},L_{2})$ are the $b_{j}$’s with $j\in{\mathcal{I}}_{\infty}$.

Note that the filtered Floer cohomology, $FH^{*}(L_{1},L_{2};-\infty,t)$
defines a persistence module equal to the one obtained from
$\operatorname{\mathcal{H}om}^{\ast}({\mathcal{F}}_{L_{1}},{\mathcal{F}}_{L_{2}})$
as follows from Theorem 5.1.

It will be useful to remind the reader that

######  Lemma A.2 (see [KS18], (1.10)).