Sheaf quantization of Legendrian isotopy

Abstract

Let $\{\Lambda ^\infty _t\}$ be an isotopy of Legendrians (possibly singular) in a unit cosphere bundle $S^*M$ that arise as slices of a singular Legendrian $\Lambda _I^\infty \subset S^*M \times T^*I$. Let $\mathcal {C}_t = Sh(M, \Lambda ^\infty _t)$ be the differential graded derived category of constructible sheaves on $M$ with singular support at infinity contained in $\Lambda ^\infty _t$. We prove that if the isotopy of Legendrians embeds into an isotopy of Liouville hypersurfaces, then the family of categories $\{\mathcal {C}_t\}$ is constant in $t$.

Motivation and results

Let $M$ be a smooth manifold, and let $Sh(M)$ be the co-complete differential graded (dg) derived category of weakly constructible sheaves on $M$ with coefficient in $\mathbb {C}$. In [KS13, Tam08, GKS12] it is proved that contact isotopy of the cosphere bundle $T^\infty M=(T^*M - T^*_M M)/\mathbb {R}_+$ acts on $Sh(M)$ as equivalences of categories. In this paper, we consider a (singular) Legendrian $\Lambda ^\infty \subset T^\infty M$ and the full subcategory $Sh(M, \Lambda ^\infty )$ consisting of sheaves $F$ with singular support at infinity, $SS^\infty (F) = (SS(F) - T^*_M M)/\mathbb {R}_+$, contained in $\Lambda ^\infty$. We define a notion of isotopy for the singular Legendrian $\Lambda ^\infty$ and prove that the category $Sh(M, \Lambda ^\infty )$ remains invariant under such an isotopy.

Such invariances of constructible sheaf categories are possible because constructible sheaves are closely related to Lagrangians in $T^*M$ (see [NZ09, GPS18a, NS20]) and hence enjoy the flexibility of symplectic geometry. More precisely, the full subcategory of compact objects in $Sh(M, \Lambda ^\infty )$, denoted by $Sh^{\rm w}(M, \Lambda ^\infty )$, is equivalent to the wrapped Fukaya category of the pair $(T^*M, \Lambda ^\infty )$ (see [GPS18a, NS20]),

\[ Sh^{\rm w}(M, \Lambda^\infty) \simeq \operatorname{Fuk}^{\rm w}(T^*M, \Lambda^\infty), \]

where the superscript ‘w’ stands for ‘wrapped’. The traditional constructible sheaves with bounded cohomologies $Sh^{pp}(M, \Lambda ^\infty )$ can be recovered as perfect modules [Nad16],

\[ Sh^{pp}(M, \Lambda^\infty) = \text{Fun}^{ex}( Sh^{\rm w}(M, \Lambda^\infty)^{op}, \operatorname{Perf}(\mathbb{C})). \]

There is an analogous result in the wrapped Fukaya category for Liouville sectors [GPS18b, Theorem 1.4]: given a Liouville domain $W$ with a ‘stop’ $S \subset \partial W$, if the contact complement $\partial W \backslash S$ remains invariant up to contact isotopy as $S$ moves, then the wrapped Fukaya category $\operatorname {Fuk}^{\rm w}(W, S)$ is invariant. Hence, combined with the comparison results of [GPS18a, NS20], we get that $Sh(M, \Lambda ^\infty )$ is invariant as long as $T^\infty M \backslash \Lambda ^\infty$ is invariant up to contact isotopy.

This paper gives a similar sufficient condition using ‘isotopy’ of $\Lambda ^\infty$: we replace $\Lambda ^\infty$ by a tube $U=U(\Lambda ^\infty )$ around the Legendrian $\Lambda ^\infty$, and we equip $U$ with a contact flow $X$ that shrinks the tube $U$ back to $\Lambda ^\infty$ (Definition 0.2). Although $\Lambda ^\infty$ may be singular, the data of $(U,X)$ are smooth, so we can talk about isotopies of $(U,X)$. The relation to the complement $T^\infty M \backslash \Lambda ^\infty$ is that if $(U_t, X_t)$ varies smoothly, then the complements $\{T^\infty M \backslash \Lambda ^\infty _t\}_t$ are contactomorphic, where $\Lambda ^\infty _t$ is the limit of $U_t$ under the shrinking flow $X_t$.

To state the main theorem precisely, we need some definitions.

Let $(C, \alpha )$ be a contact manifold with a contact 1-form $\alpha$.

Definition 0.1 A singular Legendrian $\mathcal {L} \subset C$ is a Whitney stratifiable subspace such that its top-dimensional strata are smooth Legendrians and the closure of the union of the top-dimensional strata is $\mathcal {L}$.

Definition 0.2 Let $\mathcal {L} \subset C$ be a singular Legendrian. A convex tube $(U,X)$ around $\mathcal {L}$ is an open subset $U$ containing $\mathcal {L}$ with smooth boundary $\partial U$ and a contact vector field $X$ transverse to $\partial U$ and pointing inward to $\partial U$, such that $\mathcal {L}_X(\alpha ) = -\alpha$ and $\bigcap _{u > 0} X^u(U) = \mathcal {L}$, where $X^u$ is the time-$u$ flow of $X$.

Definition 0.3 Let $I \subset \mathbb {R}$ be a closed interval and $\{(U_t, X_t, \mathcal {L}_t)\}_{t \in I}$ a family of singular Legendrians $\mathcal {L}_t$ with convex tubes $(U_t, X_t)$. If $\partial U_t$ and $X_t$ vary smoothly with $t$, we say that $\{(U_t, X_t, \mathcal {L}_t)\}_{t \in I}$ is an isotopy of convex tubes over $I$.

Let $M$ be a smooth manifold with Riemannian metric $g$. Let $S^*M \subset T^*M$ be the unit cosphere bundle, and let $\alpha = \lambda |_{S^*M}$ be a contact 1-form on $S^*M$ where $\lambda$ is the Liouville 1-form on $T^*M$ (e.g. $\lambda = p\,dx$ on $T^*\mathbb {R}$). We identify $S^*M$ with $T^\infty M$. We equip $S^*M \times T^*I$ with the contact form $\widetilde \alpha = \alpha + \tau \,dt$, where $t$ is the coordinate of $I$ and $\tau$ is the coordinate on the cotangent fiber. Then the composition $S^*M \times T^*I \hookrightarrow \dot {T}^*(M \times I) \to T^\infty (M \times I)$ is an open immersion and contactomorphism, with image $(x, t; [p, \tau ]) \in T^\infty (M \times I)$ where $p \neq 0$.

Definition 0.4 Let $I \subset \mathbb {R}$ be a closed interval. A strong isotopy of Legendrians in $S^*M$ over $I$ is a Legendrian $\mathcal {L}_I \subset S^*M \times T^*I$. A strong isotopy of convex tubes is a convex tube $(U_I, X_I)$ of $\mathcal {L}_I$ such that $X_I$ preserves the fibers of $S^*M \times T^*I \to I$.

Our main result is the following.

Theorem 0.5 If $(U_I, X_I)$ is a strong isotopy of convex tubes around $\mathcal {L}_I$ in $S^*M \times T^*I$, then for any $t \in I$ we have an equivalence of categories

\[ \iota_t^*: Sh(M \times I, \mathcal{L}_I) \to Sh(M, \mathcal{L}_t), \]

where $\iota _t: M_t = M \times \{t\} \hookrightarrow M_I = M \times I$ is the inclusion of the slice over $t$.

Given a strong isotopy of Legendrians $\mathcal {L}_I$, we prove that to construct a tube thickening $(U_I, X_I)$ it suffices to construct a Liouville hypersurface thickening of each slice $\mathcal {L}_t$ (see Proposition 1.14).

Although the result is expected given the analogous result in Fukaya category, and it is superseded by the recent paper [NS20], we hope that its purely sheaf-theoretic proof and the simpler cotangent bundle setting make the presentation of this result still worthwhile.

Previous work

We first recall the sheaf quantization of a contact isotopy of $S^*M$.

Theorem 0.6 [GKS12, Theorem 3.7 and Proposition 3.12]

Let $I$ be an open interval containing $0$, and let $\varphi : I \times T^\infty M\to T^\infty M$ be a smooth map with $\varphi _t=\varphi (t,-)$. Assume $\varphi$ is such that (i) $\varphi _0=\mathrm {id}$ and (ii) $\varphi _t$ are contactomorphisms for all $t \in I$. Then for each $t \in I$ we have the equivalences of categories

\[ \hat \varphi_t: Sh(M) \xrightarrow{\sim} Sh(M) \quad \text{such that}\quad SS^\infty(\hat \varphi_t F) = \varphi_t (SS^\infty(F)). \]

Note that any isotopy of smooth Legendrians can be extended to a contact isotopy of the ambient manifold. In general, we have the following corollary.

Corollary 0.7 If an isotopy of Legendrians $\{\Lambda ^\infty _t\}_{t \in I}$ can be embedded into an isotopy $\{\varphi _t\}_{t \in I}:S^*M \to S^*M$ of the contact manifold, that is, $\Lambda ^\infty _t = \varphi _t(\Lambda ^\infty _0)$, then we have an equivalence of categories

\[ \hat \varphi_t: Sh(M, \Lambda^\infty_0) \xrightarrow{\sim} Sh(M, \Lambda^\infty_t). \]

For a deformation of singular Legendrians, there is one necessary condition for the invariance of categories, due to Nadler [Nad15].

Definition 0.8 (Displaceable Legendrian)

Let $(S^* M, \alpha )$ be the unit cosphere bundle of a Riemannian manifold $M$ with Reeb vector field $R$ and time-$t$ Reeb flow $R^t$. A Legendrian $\mathcal {L} \subset S^* M$ is $\epsilon$-displaceable for $R$ and for some $\epsilon >0$ if

(1)\begin{equation} \mathcal{L} \cap R^s(\mathcal{L}) = \emptyset \quad \text{for all }0 < |s| < \epsilon. \end{equation}

We say that a family of Legendrians $\{\mathcal {L}_t\}$ is uniformly $\epsilon$-displaceable for $R$ and for some $\epsilon >0$ if each $\mathcal {L}_t$ is $\epsilon$-displaceable.

If a family of Legendrians $\{\mathcal {L}_t\}$ can be upgraded to an isotopy of convex tubes $\{U_t, X_t, \mathcal {L}_t\}$, then $\{\mathcal {L}_t\}$ is uniformly displaceable (Proposition 1.9).

Example 0.9 Consider the example in Figure 1.

Figure 1. The deformation to the right is uniformly displaceable, and the one to the left is not, due to the appearance of a new short Reeb chord (marked by a thick line); cf. [Nad15, Example 1.5].

The category of constructible sheaves for the three diagrams comprises the representations of the following commutative diagrams (where each region corresponds to a vertex and an arrow between vertices goes against the direction of the hair).

A sketch of the proof

Given a convex tube $(U,X)$ of a Legendrian $\mathcal {L} \subset S^*M$, we may define a projection functor as the ‘limit’ of the flow $X$:

(2)\begin{equation} \Pi_\mathcal{L}: Sh(M, U) \to Sh(M, \mathcal{L}), \quad \Pi(F) := \lim_{T \to \infty} \hat X^{T} (F), \end{equation}

where $Sh(M, U)$ stands for the category of constructible sheaves with $SS^\infty (F) \subset U$ and the limit is defined using the nearby cycle functor in § 2.6.

Let $(U_I, X_I, \mathcal {L}_I)$ be a strong isotopy of convex tubes in $S^*M \times T^*I$, and let $\{(U_t, X_t, \mathcal {L}_t)\}$ be the slices. Let $F_t \in Sh(M, \mathcal {L}_t)$. We will extend $F_t$ to a sheaf $F_I \subset Sh(M \times I, \mathcal {L}_I)$ such that $F_I|_t=F_t$.

We first show that such an extension is unique (if it exists); this is equivalent to showing that the restriction functor $Sh(M, \mathcal {L}_I) \to Sh(M, \mathcal {L}_t)$ is fully faithful (Proposition 3.1), that is,

\[ \operatorname{Hom}(F_I, G_I) \xrightarrow{\sim} \operatorname{Hom}(F_t, G_t) \quad \text{for all }F_I, G_I \in Sh(M \times I, \mathcal{L}_I). \]

One needs to show that $\mathcal {H}om(F_I, G_I)(M \times (a,b))$ is independent of the size of the interval, so that one can interpolate from $(a,b) = I$ to an infinitesimally small neighborhood around $t$. The key technical point is to use the uniform displaceability condition to perturb $G_I$ slicewise by positive Reeb flow for time $s$, $G_I \to K_s^!G_I$, to separate $SS^\infty (F_I)$ and $SS^\infty (K_s^!G_I)$.

We then show that such an extension exists locally; that is, given $F_t$, we can find a small neighborhood $J = (t-\delta, t+\delta )$ such that $\mathcal {L}_t \times T^*_J J \subset U_J = U_I \cap S^*M \times T^*J$ and extend $F_t$ on $M_t$ to $F_J$ on $M_J$ by defining $F_J = \Pi _{\mathcal {L}_J}(F_t \boxtimes \mathbb {C}_J)$.

Finally, we use the uniqueness of extension to patch together the local extensions, and thus we get the global extension result (cf. [GKS12, Lemma 1.13]).

Notation

We use $Sh(M)$ to denote the co-complete dg derived category of weakly constructible sheaves. With an abuse of notation, we use ‘constructible sheaf’ to mean a cohomologically constructible complex of sheaves. All the functors $f_*, f^*, f_!, \mathcal {H}om$ etc. are derived.

1. Convex tubes and isotopy

1.1 Basics of contact geometry

We recall the definition of co-oriented contact manifold as follows. Let $C$ be a $(2n+1)$-dimensional manifold, and let $\xi \subset TC$ be a rank-$2n$ sub-bundle such that there exists a $1$-form (contact $1$-form) $\alpha$ (up to multiplication by a non-negative function) satisfying $\xi = \ker \alpha$ and $\alpha \wedge (d\alpha )^n \neq 0$. If we fix such an $\alpha$, we have a Reeb vector field $R_\alpha$ given by

\[ \iota_{R_\alpha} \alpha = 1, \quad \iota_{R_\alpha} d\alpha = 0. \]

We remark that different choices of $\alpha$ will lead to different choices of $R_\alpha$.

A contact vector field $X$ is a vector field on $C$ that preserves the sub-bundle $\xi$.

Definition 1.1 Given a smooth function $H: C \to \mathbb {R}$, the contact Hamiltonian vector field $X_H$ is uniquely determined by the conditions

(3)\begin{equation} \begin{cases} \langle X_H, \alpha \rangle = H ,\\ \iota_{X_H} \,d \alpha = \langle dH, R \rangle \alpha - d H .\end{cases} \end{equation}

The Reeb vector field is a special case of $X_H$ where $H=1$.

Proposition 1.2 [Gei08, Theorem 2.3.1]

With a fixed choice of contact form $\alpha$, there is a one-to-one correspondence between the contact vector field $X$ and smooth functions $H: C \to \mathbb {R}$. The correspondence is given by

\[ X \mapsto H = \langle \alpha, X \rangle, \quad H \mapsto X_H. \]

Unlike symplectic Hamiltonian vector fields, $X_H$ does not preserve the level sets of $H$.

Lemma 1.3 We have that

\[ \langle X_H, dH \rangle = H \langle R, dH \rangle. \]

In particular, $X_H$ preserves the zero set of $H$.

Proof. Apply $\iota _{X_H}$ to the second line of (3) and then use the first line.

We also have the Lie derivative of $\alpha$ along $X_H$,

\[ \mathcal{L}_{X_H}(\alpha) = d \iota_{X_H} \alpha + \iota_{X_H} \,d\alpha = \langle R, H\rangle \alpha. \]

Proposition 1.4 If $\mathcal {L} \subset C$ is a germ of a smooth Legendrian and $H$ is any locally defined function vanishing on $\mathcal {L}$, then the contact flow $X_H$ is tangential to $\mathcal {L}$.

Proof. To show that $X_H$ is tangential to $\mathcal {L}$ at $p \in \mathcal {L}$, we only need to show that for any tangent vector $v \in T_p \mathcal {L}$ we have $d \alpha (X_H, v) = 0$ and $\alpha (X_H)|_p=0$, because these two conditions imply $X_H \in (T_p \mathcal {L})^{\perp _{d\alpha }} \cap \ker (\alpha )= T_p \mathcal {L}$. Indeed, $\alpha (X_H)|_p= H(p) = 0$ and

\[ d \alpha (X_H, v) = \iota_{X_H}(d\alpha)(v) = [\langle R, dH \rangle \alpha - d H](v) = \langle R, dH \rangle (\alpha(v)) - H(v)= 0. \]

Hence $X_H$ is tangential to $\mathcal {L}$.

Example 1.5 Let $M$ be a smooth manifold, and let $T^*M$ the cotangent bundle with canonical Liouville $1$-form $\lambda$ and symplectic $2$-form $\omega =d\lambda$. If we put local Darboux coordinates $(q,p)=(q_1,\ldots, q_m; p_1, \ldots, p_m)$ on $T^*M$, where $m=\dim _\mathbb {R} M$, then $\lambda = \sum _{i=1}^m p_i \,dq_i$ and $\omega = \sum _i\,dp_i \wedge dq_i$, and we will suppress the indices and summation and write simply $\lambda = p\,dq$ and $\omega = dp\,dq$. Also define $\dot {T}^*M = T^*M \backslash T^*_M M$ and $T^\infty M = \dot {T}^*M / \mathbb {R}_{>0}$. The Liouville vector field for $\lambda$ is defined by $\iota _{V_\lambda } \omega = \lambda$, and here it is given by $V_\lambda = p \partial _p$. On $T(\dot {T}^*M)$, the symplectic orthogonal to the Liouville vector field defines a distribution

\[ \widetilde \xi = \{(q,p; v_q, v_p) \in T(\dot T^*M): \omega((v_q, v_p), V_\lambda) = 0\}, \]

which projects to a canonical contact distribution $\xi$ on $T^\infty M$. Let $g$ be any Riemannian metric on $M$; then $T^*M$ has an induced norm. Let $S^*M = \{ (q, p) \in T^*M : |p|=1\}$ be the unit cosphere bundle with contact form $\alpha = \lambda |_{S^*M}$; then the contact distribution can also be written as $\xi = \ker (\alpha )$.

Define the symplectization of $(C, \xi = \ker \alpha )$ by

\[ S:= C \times \mathbb{R}_u, \quad \lambda = e^u \alpha, \quad \omega_S = d (e^u \alpha). \]

We have the projection along $\mathbb {R}_u$ and the inclusion of zero-section

\[ \pi_S: S \to C, \quad \iota_C: C \simeq C \times \{0\} \hookrightarrow S. \]

A different choice of $\alpha$ would give rise to the same $S$ up to a fiber-preserving symplectomorphism that identifies the ‘zero-section’ $\text {Im} (\iota _C)$.

A Hamiltonian function $H: C \to \mathbb {R}$ can be extended to a homogeneous degree-one function $\widetilde H: S \to \mathbb {R}$ by setting $\widetilde H =e^u H$. Then the symplectic Hamiltonian vector field $\xi _{\widetilde H}$, given by $\omega _S(-, \xi _{\widetilde H}) = d \widetilde H(-)$, preserves the fiber of $\pi _S$ and descends to $X_H$.

1.2 Convex tubes

Recall the definition of convex tubes in Definition 0.2.

Definition 1.6 A Liouville hypersurface thickening of a singular Legendrian $\mathcal {L}$ is a hypersurface $\mathcal {H} \supset \mathcal {L}$ such that $(\mathcal {H}, \alpha |_\mathcal {H})$ is a Liouville domain with the Liouville skeleton being $\mathcal {L}$.

First we show that a Liouville hypersurface thickening can be upgraded to a convex tube thickening of $\mathcal {L}$.

Proposition 1.7 Let $\mathcal {L}$ be a singular Legendrian with a Liouville hypersurface thickening $\mathcal {H}$. Then $\mathcal {L}$ admits a convex tube $(U, X)$, where the contact vector field $X$ preserves $\mathcal {H}$ and $X$ restricted to $\mathcal {H}$ is the downward Liouville flow of $\mathcal {H}$.

Proof. Let $\epsilon >0$ be small enough that for any $0< s<\epsilon$ we have $\mathcal {H} \cap R^s \mathcal {H} = \emptyset$. Then define $U = \bigcup _{|s| \leqslant \epsilon /2} R^s \mathcal {H} \simeq \mathcal {H} \times (-\epsilon /2, +\epsilon /2)$, and let $h: U \to (-\epsilon /2, +\epsilon /2)$ be the projection. Then $X = X_h$ shrinks $U$ to $\mathcal {L}$ and restricts to the downward Liouville flow on $\mathcal {H}$. One may smooth the corner of $U$ and achieve transversality of $X$ with $\partial U$.

Conversely, we show that each convex tube $(U,X)$ around $\mathcal {L}$ determines a Liouville thickening.

Proposition 1.8 Let $(U,X)$ be a convex tube around $\mathcal {L}$. Let $h=\alpha (X)$ and $\mathcal {H}= h^{-1}(0) \subset U$. Then $\mathcal {H}$ is a Liouville thickening of $\mathcal {L}$.

Proof. Since $X = X_h$ preserves $\mathcal {H}$ and shrinks $\mathcal {H}$ to $\mathcal {L}$, we only need to show that $\mathcal {H}$ is transverse to $R$ and $X$ is the downward Liouville flow on $\mathcal {H}$.

Since $\mathcal {L}_X(\alpha )=\langle R, dh \rangle \alpha = -\alpha$, we have $R(h)=-1$. Thus $R$ is transversal to the level sets of $h$, in particular $\mathcal {H}$. Hence $d\alpha$ is non-degenerate on $\mathcal {H}$, so $\mathcal {H}$ is exact symplectic. Let $\lambda = \alpha |_\mathcal {H}, \omega = d\lambda$. When we restrict to $T\mathcal {H}$, we have

\[ \iota_{X_h} (\omega) =\iota_{X_h}( d\alpha) = \langle R, dh \rangle \alpha - dh = - \lambda, \]

and hence $X_h$ is the downward Liouville flow on $\mathcal {H}$.

Proposition 1.9 Let $(U,X)$ be a convex tube of $\mathcal {L}$. Then $\mathcal {L}$ is displaceable (see Definition 0.8). Similarly, let $I=[0,1]$ and let $(U_I, X_I)$ be a strong isotopy of convex tubes for $\mathcal {L}_I$. Then the family of Legendrians $\mathcal {L}_t$ are uniformly displaceable.

Proof. Let $h =\alpha (X)$ be the Hamiltonian function generating $X$. Then $h$ vanishes on $\mathcal {L}$, and by the normalization condition we have $R(h)=-1$. If there is a Reeb chord $\gamma : [0, T] \to C$ contained in $U$ and ending on $\mathcal {L}$, then we have

\[ \int_0^T \dot \gamma (dh) \,dt = \int_0^T R (dh) \,dt = \int_0^T (-1) \,dt = -T. \]

But on the other hand we also have

\[ \int_0^T \dot{\gamma}(dh) \,dt = \int_\gamma \,dh = h(\gamma(T)) - h(\gamma(0)) = 0, \]

since $\gamma (T) \in \mathcal {L}$, $\gamma (0) \in \mathcal {L}$ and $h|_\mathcal {L} = 0$. Thus, there is no Reeb chord ending on $\mathcal {L}$ and contained in $U$. For any $x \in \mathcal {L}$, let $t(x) = \inf \{t \in \mathbb {R}: R^t(x) \notin U\}$; then $t(x)>0$ and is continuous on $\mathcal {L}$. Let $\epsilon =\inf \{t(x): x \in \mathcal {L}\}$; since $\mathcal {L}$ is compact, $\epsilon >0$. Then $\mathcal {L}$ is displaceable.

For the uniformly displaceable statement, note that $I=[0,1]$ is compact and $\epsilon (t)$ for $(U_t,X_t)$ is continuous in $t$; hence $\epsilon = \inf \{\epsilon (t)\}>0$.

1.3 The construction of strong isotopies of convex tubes

Consider the unit cosphere bundle $(S^*M, \alpha )$ and a closed interval $I \subset \mathbb {R}$. Let a point in $S^*M$ be denoted by $(x,p) \in T^*M$ with $|p| = 1$. Let a point in $T^*I$ be denoted by $(t, \tau ) \in I \times \mathbb {R}$. Let $S^*M \times T^*I$ be equipped with the contact 1-form

\[ \alpha_I = \alpha + \tau \,d t. \]

Let $\pi _t: S^*M \times T^*I \to I$.

Proposition 1.10 The Reeb flow $R_I$ on $S^*M \times T^*I$ for $\alpha _I$ is the pullback of the Reeb flow $R$ on $S^*M$.

Proof. Let $R$ denote the pullback to $S^*M \times T^*I$. We may verify that $\iota _R(\alpha _I)=1$ and $\iota _R(d \alpha _I) = 0$.

Let $\mathcal {L}_I$ be a strong isotopy of Legendrians. Let

\[ \mathcal{L}_t = \{ (x,p) \in S^*M \mid \exists (x,p, t, \tau) \in \mathcal{L}_I\}. \]

Lemma 1.11 We have that $\mathcal {L}_t$ is a singular Legendrian in $S^*M$.

Proof. Take any $p \in \mathcal {L}_t$ that is the image of a point $\widetilde p$ in the smooth loci $\mathcal {L}_I^{sm}$; any tangent vector $v \in T_p \mathcal {L}_t$ can be lifted to $\widetilde v \in T_{\widetilde p} \mathcal {L}_I$. Concretely, $\widetilde v = v + c \partial _\tau$. Since $0 = (\alpha + \tau \, dt) (\widetilde v) = \alpha (v)$, we see that $T_p \mathcal {L}_t$ is in $\ker (\alpha )$. Hence a dense open part of $\mathcal {L}_t$ is Legendrian, and thus $\mathcal {L}_t$ is a singular Legendrian.

Let $(U_I, X_I, \mathcal {L}_I)$ be a strong isotopy of convex tubes. First we define the restriction to $S^*M \times T_t^* I$. Since $X_I$ preserves the $t$ coordinate, for each $t$ we have the vector field

\[ \hat{X}_t := X_I|_t \in \operatorname{Vect} (S^*M \times T_t^* I). \]

Also define the restriction

\[ \hat U_t = U_I \cap S^*M \times T^*_t I, \quad \hat{\mathcal{L}}_t = \mathcal{L}_I \cap S^*M \times T^*_t I. \]

Next, we define $(U_t, X_t)$. Define the projection map $\hat \pi _t: S^*M \times T^*_t I \to S^*M$ and write

\[ U_t = \hat \pi_t(\hat U_t), \quad \mathcal{L}_t = \hat \pi_t (\hat{\mathcal{L}}_t). \]

Let $h_I = \alpha _I(X_I)$; since $X_I$ has no $\partial _t$ component, $\partial _\tau h_I=0$ and hence $h_I$ is independent of $\tau$. For each $t \in I$ we define

\[ h_t(x,p) := h_I(x,p,t)\quad \text{for all }(x,p) \in U_t, \]

and we let $X_t$ be the contact vector field generated by $h_t$.

Proposition 1.12 With the above setup, we have

\[ \hat{X}_t = X_t + (-\tau - \partial_t h_t) \partial_\tau. \]

Proof. We split a tangent vector $v$ on $S^*M \times T^*I$ into two components as $v = v_1 + v_2$, where $v_1$ and $v_2$ are along the $S^*M$ and $T^*I$ factors, respectively. Similarly, we decompose $X_I$ as $X_I = X_{I,1} + X_{I,2}$, where $X_{I,2} = a \,\partial _\tau$.

By the definition of $X_I$, we have

\[ \iota_{X_I} (\alpha + \tau \,dt) = h_I \]

and

\[ \iota_{X_I} \,d (\alpha + \tau \,dt) = \langle R_I, h_I \rangle (\alpha + \tau \,dt) - d h_I, \]

which we will refer to as the first and second equations below.

Since $\tau \,dt(X_{I,2}) = 0$, the first equation becomes

\[ \alpha(X_{I,1}) = h_t(x, p). \]

For the second equation, if we restrict to the tangent space on $S^*M$, we have

\[ \iota_{X_{I,1}} \,d\alpha = \langle R, h_t \rangle \alpha - d h_t. \]

Thus $X_{I,1} = X_t$ is the contact vector field on $S^*M$ generated by $h_t(x,p)$.

Finally, if we restrict the second equation to the tangent space of $T^*I$, we get

\[ \iota_{X_{I,2}}(d\tau \wedge dt) = \langle R, h_I \rangle (\tau\, dt) - \partial_t h_t \,dt. \]

If we plug in $X_{I,2} = a \partial _t$ and $\langle R, h_I \rangle = -1$, we get the desired result.

Proposition 1.13 For any $t \in I$, the $(U_t, X_t)$ defined above is a convex tube for $\mathcal {L}_t$. Furthermore, the family $\{(U_t, X_t)\}_t$ varies smoothly with $t$ and hence is an isotopy of convex tubes for $\{\mathcal {L}_t\}$.

Proof. From Proposition 1.12 we know that the flow of $\hat {X}_t$ preserves the fibers of $S^*M \times T_t^* I \to S^*M$ and the induced flow on $S^*M$ is generated by $X_t$. Since the flow of $\hat {X}_t$ shrinks $\hat U_t$ to $\hat {\mathcal {L}}_t$, i.e. $\hat {\mathcal {L}}_t = \bigcap _{u > 0} \hat {X}_t^u \hat U_t$, the sequence of open sets $\hat {X}_t^u \hat U_t$ is monotonically decreasing in $u$; furthermore, $\hat \pi _t(\hat {X}_t^u \hat U_t) = X_t^u (U_t)$, so we have

\[ \mathcal{L}_t = \bigcap_{u>0} X_t^u (U_t).\]

The final proposition allows us to upgrade from an isotopy of Liouville hypersurfaces to a strong isotopy of convex tubes.

Proposition 1.14 If $\mathcal {L}_I$ is a Legendrian in $S^*M \times T^*I$ and if $\{\mathcal {H}_t\}$ is a smooth family of Liouville hypersurfaces in $S^*M$ such that $\mathcal {L}_t$ is the skeleton of $\mathcal {H}_t$, then we have a strong isotopy of convex tubes $(U_I, X_I)$ around $\mathcal {L}_I$.

Proof. First we use $\mathcal {H}_t$ to get a family of convex tubes $(U_t, X_t)$ and the associated Hamiltonian functions $h_t$, where $h_t|_{\mathcal {H}_t}=0$ and $R(h_t)=-1$. The family of functions $h_t$ determines the lifted function $h_I(x,p,t,\tau ) = h_t(x,p)$ which is defined when $(x,p) \in U_t$. In turn, $h_I$ determines the contact vector field $X_I$, which restricts to the fiber $S^*M \times T^*_t I$ as given by Proposition 1.12. Thus, we only need to specify the subset $\hat U_t \subset U_t \times T^*_t I$ such that its boundary $\partial \hat U_t$ is transverse to the vector field $\hat X_t$ and it is compressed by the flow of $\hat X_t$ to $\hat {\mathcal {L}}_t = \mathcal {L}_I \cap S^*M \times T^*_t I$.

Let

\[ C = 1 + \sup \{ |\partial_t h_t(x,p)| : (x,p) \in \bar{U}_t, \, t \in I \} \]

and let

\[ \hat U_t = U_t \times (-C, C) \subset S^*M \times T^*I. \]

Then the flow $\hat X_t$ is transverse to the boundary $\partial U_t$. We only need to show that $\bigcap _{u>0} \hat X_t^u(\hat U_t) = \hat {\mathcal {L}}_t$. Since $\hat U_t \to U_t$ with fiber $(-C, C)$, $\hat X_t$ restricted to the fiber gives the equation for $\tau$ as

\[ (d/du) \tau(u) = - \tau - \partial_t h_t(x(u), p(u)). \]

This is a contracting flow with unit contraction rate in the sense that for any initial condition $(\tau _1, \tau _2)$ at $u=0$ we have $\tau _1(u)-\tau _2(u) = (\tau _1 - \tau _2) e^{-u}$ for $u>0$.

Under the projection map $\hat \pi _t: S^*M \times T_t^*I \to S^*M$, we have the surjection

\[ \hat \pi_t: \hat{\mathcal{L}}'_t :=\bigcap_{u>0} \hat X_t^u \hat U_t \to \bigcap_{u>0} X_t^u U_t = \mathcal{L}_t, \]

and by the contracting property of the flow $\hat X_t$, the fiber can consist of only one point; thus $\hat \pi _t: \hat {\mathcal {L}}'_t \to \mathcal {L}_t$ is a bijection.

Let $U_I = \bigcup _{t \in I} \hat U_t \subset S^*M \times T^*I$, and put the slices $\hat {\mathcal {L}}'_t$ together into $\mathcal {L}'_I = \bigcap _{u > 0} X_I^u U_I$. Recall $\hat \pi : S^*M \times T^*I \to S^*M \times I$; then $\hat \pi (\mathcal {L}_I) = \hat \pi (\mathcal {L}'_I)$ and $\mathcal {L}'_I$ is homeomorphic to its image. Since a smooth family of smooth Legendrians in $S^*M \times I$ has a unique lift to $S^*M \times T^*I$, $\:\mathcal {L}_I$ and $\mathcal {L}'_I$ agree over the smooth loci of $\hat \pi (\mathcal {L}_I)$. Since $\mathcal {L}_I$ is the closure of its smooth part, we have $\mathcal {L}_I = \mathcal {L}'_I$, finishing the proof of the proposition.

2. Non-characteristic isotopy of sheaves

2.1 Constructible sheaves

We give a quick working definition of constructible sheaves used in this paper, and refer to [KS13] for a proper treatment. A constructible sheaf $F$ on $M$ is a sheaf valued in a chain complex of $\mathbb {C}$-vector spaces, such that its cohomology is locally constant with finite rank with respect to some Whitney stratification¹ $\mathcal {S} = \{\mathcal {S}_\alpha \}_{\alpha \in A}$ on $M$, where the $\mathcal {S}_\alpha$ are disjoint locally closed smooth submanifolds with a nice adjacency condition and $M= \bigsqcup _{\alpha \in A} \mathcal {S}_\alpha$. The singular support $SS(F)$ of $F$ is a closed conical Lagrangian in $T^*M$, contained in $\bigcup _{\alpha \in A} T_{\mathcal {S}_\alpha }^* M$, such that $SS(F) \cap T^*_M M$ equals the support of $F$ and $(p,q) \in SS(F) \backslash T^*_M M$ if there exists a locally defined function $f$ with $f(q)=0$ and $df(q)=p$ such that the restriction map $F(B_\epsilon (q) \cap \{f<\delta \}) \to F(B_\epsilon (q) \cap \{f<-\delta \})$ fails to be a quasi-isomorphism for $0 < \delta \ll \epsilon \ll 1$. We denote by $SS^\infty (F) = SS(F) \cap S^*M$ the singular support of $F$ at infinity.

If $\Lambda \subset T^*M$ is a conical Lagrangian containing the zero-section (as assumed throughout this paper), we write $Sh(M, \Lambda ^\infty )$ for the dg derived category of constructible sheaves with object $F$ satisfying $SS^\infty (F) \subset \Lambda ^\infty$.

Example 2.1 On $\mathbb {R}$, let $\mathbb {C}_{[0,1]}$ (respectively $\mathbb {C}_{(0,1)}$) denote the constant sheaf with stalk $\mathbb {C}$ on $[0,1]$ (respectively on $(0,1)$) and zero stalk elsewhere. Then the singular supports of $\mathbb {C}_{[0,1]}$ and $\mathbb {C}_{(0,1)}$ in $T^*\mathbb {R}$ are

Example 2.2 Let $j: U=B(0,1) \hookrightarrow \mathbb {R}^2$ be the inclusion of an open unit ball in $\mathbb {R}^2$. Then $j_* \mathbb {C}_U$ is supported on the closed set $\overline U$, with singular support at infinity as

And $j_! \mathbb {C}_U$ is supported on the open set $U$, with singular support at infinity given by

Here the Legendrians are represented by co-oriented hypersurfaces in $\mathbb {R}^2$ with hairs indicating the co-orientation.

2.2 Operation on sheaves

In this subsection, we deviate from our running convention and use $Sh(X)$ to denote the co-complete dg derived category of sheaves on $X$ without any constructibility condition. Let $f: Y \to X$ be a map of real analytic manifolds. Then we have the following pairs of adjoint functors:

\begin{align*} - \otimes F&{:}\, Sh(X) \leftrightarrow Sh(X) {:}\, \mathcal{H}om(F, -),\\ f^*&{:}\, Sh(X) \leftrightarrow Sh(Y) {:}\, f_* ,\\ f_!&{:}\, Sh(Y) \leftrightarrow Sh(X){:}\, f^!. \end{align*}

Given an open subset $U$ of $X$ and its closed complement $Z$,

\[ \text{open inclusion:} \quad U {\buildrel j \over \hookrightarrow} X {\buildrel i \over \hookleftarrow} Z, \quad \text{closed inclusion}, \]

we have $j^* = j^!$ and $i_*=i_!$. Furthermore, there are exact triangles

\[ i_! i^! \to \mathrm{id} \to j_* j^* \xrightarrow{[1]}, \quad j_! j^! \to \mathrm{id} \to i_* i^* \xrightarrow{[1]}. \]

These are sheaf-theoretic incarnations of excisions: upon applying to the constant sheaf on $X$ and taking global sections, we get

\[ H^*(Z, i^! \mathbb{C}) \to H^*(X, \mathbb{C}) \to H^*(U, \mathbb{C}) \xrightarrow{[1]}, \quad H^*_c(U, \mathbb{C}) \to H^*_c(X, \mathbb{C}) \to H^*_c(Z, \mathbb{C}) \xrightarrow{[1]}. \]

Let $X_i$, $i=1,2$, be spaces, and let $K \in Sh(X_1 \times X_2)$. We define the pair of adjoint functors

(4)\begin{gather} K_!: Sh(X_1) \leftrightarrow Sh(X_2): K^! ,\end{gather}

(5)\begin{gather} K_!: F \mapsto {\pi_2}_! (K \otimes \pi_1^* F), \quad K^!: G \mapsto {\pi_1}_*(\mathcal{H}om(K, \pi_2^! G)). \end{gather}

In [KS13], $K_! = \Phi _K$ and $K^!=\Psi _K$ with $X_1$ and $X_2$ switched. The notation here is suggestive of their being adjoint functors.

2.3 Isotopy of constructible sheaves

Let $I =(a,b) \subset \mathbb {R}$. For any $t \in I$, let

\[ j_t: M_t := M \times \{t\} \hookrightarrow M_I := M \times I \]

be the inclusion of the $t$-slice $M_t$ into the total space $M_I$, and let $\pi _I: M_I \to I$ be the projection. Let $\mathbb {C}_{M_t}$ be the constant sheaf on $M_t$ with stalk $\mathbb {C}$. We then have

\[ SS(\mathbb{C}_{M_t}) = \{ (x, t; 0, \tau) \in T^* M_I \}, \quad SS^\infty(\mathbb{C}_{M_t}) = \{ (x, t; 0, \pm 1) \in S^* M_I \simeq T^\infty M\}. \]

Definition 2.3 Let $M$ be a smooth manifold and $I$ a closed interval of $\mathbb {R}$.

1. (i) An isotopy of (constructible) sheaves is a constructible sheaf $F_I \in Sh(M \times I)$ such that $SS^\infty (F_I)$ is a strong isotopy of Legendrians in $S^*M \times T^* I$ (Definition 0.4). Equivalently, for any $t \in I$ we have

   \[ SS^\infty(F_I) \cap SS^\infty(\mathbb{C}_{M_t}) = \emptyset. \]
   If $F_I$ is an isotopy of sheaves, then for any $t \in I$ we denote the restriction of $F_I$ at $t$ by
   \[ F_t: = F_I|_{M_t} \in Sh(M). \]

2. (ii) Two isotopies of sheaves $F_I, G_I \in Sh(M \times I)$ are said to be non-characteristic if

   \[ SS^\infty(F_I)|_t \cap SS^\infty(G_I)|_t = \emptyset \quad \text{for all }t \in I. \]

Some easy-to-check properties are the following.

Proposition 2.4 Let $M$ be a compact real analytic manifold.

1. (1) If $F_I$ is an isotopy of sheaves and $\Lambda ^\infty _I = SS^\infty (F_I)$, then

   \[ SS^\infty(F_t) \subset \Lambda^\infty_t. \]

2. (2) If $F_I$ is an isotopy of sheaves and $\pi _I: M_I \to I$, then $(\pi _I)_* F_I$ is a local system on $I$.

2.4 Invariance of morphisms under non-characteristic isotopies

We use the same notation for $M_I=M \times I, \,M_t, \,\mathbb {C}_{M_t}$ etc. as in the previous subsection.

Lemma 2.5 Let $F \in Sh(M)$ and let $\varphi : M \to \mathbb {R}$ be a $C^1$ function such that $d\varphi (x)\neq 0$ for $x \in \varphi ^{-1}([0,1])$.

1. (1) For $s \in (0,1)$ let $U_s = \{x: \varphi (x) < s \}$, and let $U_1 = \bigcup _{s} U_s$. If

   \[ SS^\infty(\mathbb{C}_{U_s}) \cap SS^\infty(F) = \emptyset \quad \text{for all } 0< s<1, \]
   then
   \[ \operatorname{Hom}(\mathbb{C}_{U_1}, F) \xrightarrow{\sim} \operatorname{Hom}(\mathbb{C}_{U_s}, F)\quad \text{for all } 0< s<1. \]

2. (2) For $s \in (0,1)$ let $Z_s = \{x: \varphi (x) \leqslant s \}$, and let $Z_0 = \bigcap _{s} Z_s$. If

   \[ SS^\infty(\mathbb{C}_{Z_s}) \cap SS^\infty(F) = \emptyset\quad \text{for all } 0< s<1, \]
   then
   \[ \operatorname{Hom}(\mathbb{C}_{Z_s}, F) \xrightarrow{\sim} \operatorname{Hom}(\mathbb{C}_{Z_0}, F)\quad \text{for all } 0< s<1. \]

Proof. Assertion (1) is a special case of [GKS12, Proposition 1.8]. Assertion (2) follows from (1) and the fact that

\[ 0 \to \mathbb{C}_{M \backslash \mathbb{Z}_s} \to \mathbb{C}_M \to \mathbb{C}_{Z_s} \to 0. \]

The following lemma is also often used.

Lemma 2.6 (Petrowsky theorem for sheaves [KS13])

Let $F, G \in Sh(M)$ be (cohomologically) constructible complexes of sheaves. If $SS^\infty (F) \cap SS^\infty (G) = \emptyset$, then the natural morphism

\[ \mathcal{H}om(F, \mathbb{C}_M) \otimes G \to \mathcal{H}om(F, G) \]

is an isomorphism.

Corollary 2.7 If $F_I$ is an isotopy of sheaves, then

\[ \mathcal{H}om(\mathbb{C}_{M_t}, F_I) \simeq \mathbb{C}_{M_t}[-1] \otimes F_I. \]

Proposition 2.8 Let $G_I$ and $F_I$ be non-characteristic isotopies of sheaves; then $\mathcal {H}om(F_I, G_I)$ is an isotopy of sheaves. In particular,

\[ \operatorname{Hom}(F_t, G_t) \simeq \operatorname{Hom}(F_s, G_s) \quad \text{for all } t, s \in I. \]

Proof. As $G_I$ and $F_I$ being non-characteristic implies $SS^\infty (G_I) \cap SS^\infty (F_I) = \emptyset$, we can bound the singular support of the hom-sheaf as [KS13]

\[ SS(\mathcal{H}om(F_I, G_I)) \subset SS(G_I) + SS(F_I)^a. \]

Again, using that $G_I$ and $F_I$ are non-characteristic, we obtain

\[ SS^\infty (\mathcal{H}om(F_I, G_I)) \cap SS^\infty(\mathbb{C}_{M_t}) = \emptyset \quad \text{for all } t, s \in I. \]

Hence $\mathcal {H}om(F_I, G_I)$ is an isotopy of sheaves. For the second statement, we have

(6)\begin{align} \operatorname{Hom}(F_t, G_t) &= \operatorname{Hom}({j_t}^* F_I, {j_t}^* G_I) \simeq \operatorname{Hom}(F_I, {j_t}_* j_t^* G_I) \simeq \operatorname{Hom}(F_I, \mathbb{C}_{M_t} \otimes G_I) \nonumber\\ &\simeq \operatorname{Hom}(F_I, \mathcal{H}om(\mathbb{C}_{M_t}, G_I)[1]) \simeq \operatorname{Hom}(\mathbb{C}_{M_t}, \mathcal{H}om(F_I, G_I))[1] \nonumber\\ &\simeq \operatorname{Hom}(\mathbb{C}_t, \pi_{I*} \mathcal{H}om(F_I, G_I))[1] \simeq [\pi_{I*} \mathcal{H}om(F_I, G_I)]_t . \end{align}

The result then follows since $\pi _{I*}(\mathcal {H}om(F_I, G_I))$ is a local system.

2.5 Invariance of morphisms under Reeb perturbations

Sometimes we want to vary $G$ and $F$ while preserving $\operatorname {Hom}(F,G)$, but $SS^\infty (G) \cap SS^\infty (F) \neq \emptyset$, e.g. $F=G$. Here we borrow an idea from the infinitesimally wrapped Fukaya category [NZ09], namely that to compute $\operatorname {Hom}_{Fuk}( L_1, L_2)$ one needs to perform a perturbation to separate $L_1$ and $L_2$ at infinity; one can perturb $L_2 \leadsto R^t L_2$ or $L_1 \leadsto R^{-t} L_1$, where $R^t$ is the unit-speed geodesic flow on $T^*M$ (smoothed near the zero-section) for positive small times $t$, small enough that no new intersections are created between $L_1$ and $L_2$ at infinity.

Fix a Riemannian metric $g$ on $M$ and identify $S^*M$ with $T^\infty M$, so that the Reeb flow $R^t$ is the unit-speed geodesic flow. Let $r_{\rm inj}(M, g)$ be the injective radius of $(M,g)$. Let $\hat R^t$ be the GKS quantization of $R^t$. The rest of this subsection will be devoted to proving the following proposition.

Proposition 2.9 Let $\Lambda ^\infty \subset T^\infty M$ be a Legendrian, and let $0 < \epsilon < r_{\rm inj}(M, g)$ be small enough that

\[ \Lambda^\infty \cap R^t \Lambda^\infty = \emptyset \quad \text{for all } 0 < |t| < \epsilon. \]

1. (1) For any $F \in Sh(M, \Lambda )$ and $0\leqslant t<\epsilon$, we have a canonical morphism

   \[ F \to \hat R^t F. \]

2. (2) For any $F,G \in Sh(M, \Lambda )$ and $0 \leqslant t<\epsilon$, we have canonical quasi-isomorphisms

   \[ \operatorname{Hom}(F, G) \xrightarrow{\sim} \operatorname{Hom}(F, \hat R^t G), \quad \operatorname{Hom}(F, G) \xrightarrow{\sim} \operatorname{Hom}(\hat R^{-t}F, G). \]

Proof. For any $0\leqslant t < \epsilon$, define

\[ K_t=\mathbb{C}_{\{(x,y) | d_g(x,y) \leqslant t\}} \in Sh(M \times M). \]

Then, from [GKS12], we have

\[ \hat R^t F = \pi_{1*} \mathcal{H}om(K_t, \pi_2^! F) = K_t^! F \]

and

\[ \hat R^{-t} F = \pi_{2!} (K_t \otimes \pi_1^* F) = (K_t)_! F, \]

where $\pi _1$ and $\pi _2$ are the projections from $M \times M$ to the first and second factors, and $\mathcal {H}om$ is the (dg derived) sheaf-hom. From the canonical restriction morphism $K_t \to K_0=\mathbb {C}_\Delta$, where $\Delta \subset M \times M$ is the diagonal subset, we have

\[ F = \pi_{1*} \mathcal{H}om(K_0, \pi_2^! F) \to \pi_{1*} \mathcal{H}om(K_t, \pi_2^! F) = \hat R^t F. \]

For statement (2) of the proposition, we first prove the following lemma.

Lemma 2.10 We have that

(7)\begin{equation} SS^\infty (K_t) \cap SS^\infty(\mathcal{H}om(\pi_1^*F, \pi_2^! G)) = \emptyset\quad \text{for all } 0 < t < \epsilon . \end{equation}

Proof. We identify the contact infinity $T^\infty M$ with the unit cosphere bundle $S^*M$. Assume that the intersection is non-empty and contains the point $(x_1, x_2, p_1, p_2)$. Since $(x_1, x_2; p_1, p_2) \in SS^\infty (K_t)$, we have

\[ d_g(x_1, x_2) = t. \]

Since $t < \epsilon < r_{\rm inj}(M,g)$, there is a unique length-$t$ geodesic $\gamma$ connecting $x_1$ and $x_2$, and $p_i$ is the unit tangent vector along $\gamma$ at $x_i$ pointing to the interior of the geodesic,

\[ p_i = -\partial_{x_i}d_g(x_1, x_2). \]

Hence the geodesic flow on $S^*M$ relates $(x_i, p_i)$ via

(8)\begin{equation} R^t(x_1, p_1) = (x_2, -p_2), \quad R^t( x_2, p_2)= (x_1, -p_1). \end{equation}

On the other hand, since $(x_1, x_2; p_1, p_2) \in SS^\infty (\mathcal {H}om(\pi _1^*F, \pi _2^! G))$, we have

(9)\begin{equation} (x_1, -p_1) \in SS^\infty (F), \quad (x_2, p_2)\in SS^\infty(G). \end{equation}

Hence, combining (8) and (9), we have

\[ (x_1, -p_1) \in R^t( SS^\infty(G)) \cap SS^\infty(F) \subset R^t \Lambda^\infty \cap \Lambda^\infty. \]

This contradicts the displaceability of $\Lambda ^\infty$ for $t < \epsilon$.

Now we come back to the proof of Proposition 2.9. We have

\begin{align*} \operatorname{Hom}(F, G) &\simeq \Gamma(M, \mathcal{H}om(F,G)) \\ &\simeq \Gamma(M \times M, \mathcal{H}om(\mathbb{C}_\Delta, \mathcal{H}om(\pi_1^*F,\pi_2^!G))) \\ &\xrightarrow{\sim} \Gamma(M \times M, \mathcal{H}om(K_t, \mathcal{H}om(\pi_1^*F,\pi_2^!G))) \\ &\simeq \Gamma(M \times M, \mathcal{H}om(\pi_1^*F, \mathcal{H}om(K_t,\pi_2^!G))) \\ &\simeq \Gamma(M, \mathcal{H}om(F, \pi_{1*}\mathcal{H}om(K_t,\pi_2^!G))) \\ &\simeq \operatorname{Hom}(F, \hat R^t G), \end{align*}

where in the third step we used the canonical morphism $K_t \to \mathbb {C}_\Delta$ when replacing $\mathbb {C}_\Delta$ by $K_t$, and used Lemmas 2.10 and 2.5(2) to show that it is a quasi-isomorphism.

We will use the following purely sheaf-theoretic statement later to study the family of GKS quantization.

Proposition 2.11 Let $I=(0,1)$, and let $K_I \in Sh(M \times M \times I)$ be an isotopy of sheaves such that $K_t = \mathbb {C}_{\Delta _t}$ for some closed subsets $\{\Delta _t\}_{0< t<1}$ satisfying

\[ \Delta_t \subset \Delta_s \;\;\forall\, 0< t< s<1 \quad \text{and}\quad \bigcap_{t \in I} \Delta_t = \Delta_M = \{(x,x): x \in M\}. \]

Let $F, G \in Sh(M, \Lambda )$, and let $\mathcal {H}om(\pi _1^*F,\pi _2^!G) \in Sh(M \times M)$ be the hom-sheaf. Assume that

\[ SS^\infty(K_t) \cap SS^\infty(\mathcal{H}om(\pi_1^*F,\pi_2^!G)) = \emptyset\quad \text{for all }t \in I; \]

then

\[ \operatorname{Hom}(F,G) \simeq \operatorname{Hom}(F, K_t^! G) \simeq \operatorname{Hom}(K_{t!} F, G) \quad \text{for all }t \in I, \]

where $K_t^!, K_{t!}$ are defined in (5).

The proof is exactly the same as that of Proposition 2.9(2), where the condition provided in Lemma 2.10 is put into the hypothesis, so we do not repeat it here.

2.6 Limit of contact isotopy

Let $I=(0,1)$ and define the inclusions

\[ (0,1) {\buildrel {j_I} \over \hookrightarrow} \mathbb{R} {\buildrel {j_0} \over \hookleftarrow} \{0\}. \]

Proposition 2.12 [TWZ19, Lemma 7.1]

Let $F_I \in Sh(M_I)$ be an isotopy of constructible sheaves, and let $\Lambda ^\infty _I = SS^\infty (F_I)$. Suppose the family ${(\Lambda _t^\infty, t)} \subset T^\infty M \times (0,1)$ has a closure in $T^\infty M \times [0,1)$ whose intersection with $T^\infty M \times \{0\}$ is a Legendrian $\Lambda _0^\infty$. Then the sheaf

(10)\begin{equation} F_0: = (j_0)^*(j_I)_* F_I \end{equation}

has $SS^\infty (F_0) \subset \Lambda ^\infty _0$.

Proof. These are corollaries of results in [KS13]. By [KS13, Theorem 6.3.1], a point $(x,p; 0, -1) \in \dot T^*M \times T^*\mathbb {R}$ belongs to $SS((j_I)_* F_I)$ only if $(x,p)$ is the limit of a sequence of points $(x_n, p_n) \in \Lambda _{t_n}$ where $t_n \to 0$, i.e. $(x,p) \in \Lambda _0$. By [KS13, Proposition 5.4.5], $SS(F_0) \subset SS((j_I)_* F_I)|_0 = \Lambda _0$; hence $SS^\infty (F_0) \subset \Lambda _0^\infty$.

Let $(U,X)$ be a convex tube for a Legendrian $\mathcal {L} \subset S^*M$. Let $X$ be extended from a neighborhood of $\bar U$ to all of $S^*M$. Let

\[ \hat X^{[0,\infty)}: Sh(M) \to Sh(M \times [0,\infty)) \]

be the sheaf quantization of the flow $X$, and let

\[ j_{[0,\infty)}: \ [0,\infty) \hookrightarrow \ [0,\infty]\ \hookleftarrow \{\infty\}: \ j_{\infty}. \]

Then we define the functor $\Pi _X: = ({\rm id}_M \times j_\infty )^* \circ ({\rm id}_M \times j_{[0,\infty )})_* \circ \hat X^{[0,\infty )}: Sh(M) \to Sh(M).$

Let $Sh(M, U)$ denote the subcategory of $Sh(M)$ consisting of sheaves $F$ with $SS^\infty (F) \subset U$.

Proposition 2.13 When restricted to $Sh(M, U)$, we have that

\[ \Pi _{U, X} = \Pi _X|_{Sh(M, U)}: Sh(M, U) \to Sh(M, \mathcal {L}). \]

Proof. This follows from the definition of a convex tube and Proposition 2.12.

3. Existence and uniqueness of the extension

In this section we prove our main result, Theorem 0.5. In this section, we will sometimes identify $\Lambda ^\infty _t \subset T^\infty M$ with $\mathcal {L}_t \subset S^*M$ and identify Reeb flow with geodesic flow.

3.1 Uniqueness of extension

Recall from Proposition 1.9 that existence of strong isotopy of convex tubes implies uniform displaceability of the family $\{\mathcal {L}_t\}$.

Proposition 3.1 Let $\Lambda ^\infty _t$ be a family of Legendrians in $T^\infty M$ that are uniformly displaceable with parameter $\epsilon$. Then the restriction functor $\iota _t^*$ is fully faithful for all $t$.

Proof. For $0 \leqslant s < \epsilon$ we define a family of kernels in $Sh((M_1 \times I_1) \times (M_2 \times I_2))$ as

(11)\begin{equation} K_s := \mathbb{C}_{d(x_1, x_2) \leqslant s} \boxtimes \mathbb{C}_{t_1=t_2}. \end{equation}

One can check that $K_s$ generates the slicewise geodesic flow, i.e. if $F_I \in Sh(M_I)$ and

\[ K_s^! F_I:= \pi_{1*} \mathcal{H}om(K_s, \pi_2^! F_I), \]

then

\[ SS^\infty( (K_s^! F_I)|_{M_t} ) = R^s SS^\infty(F_I|_{M_t}), \]

where $\pi _i$ is the projection from $(M_1 \times I_1) \times (M_2 \times I_2)$ to $M_i \times I_i$ and $R^s$ is the Reeb (geodesic) flow for time $s$.

We first prove the following claim: for any $F_I, G_I \in Sh(M_I, \Lambda ^\infty _I)$,

\[ \operatorname{Hom}(\mathbb{C}_{M \times (a,b)}, \mathcal{H}om(F_I, G_I)) \ \text{is independent of $(a,b) \subset I$. } \]

It suffices to prove the case for the right endpoint $b$. To use the estimate of the singular support of the hom-sheaf, we would like to perturb $G_I$ by the fiberwise Reeb flow.

Lemma 3.2 For any $0< s<\epsilon$,

\[ \operatorname{Hom}(\mathbb{C}_{M \times \{t\}}, \mathcal{H}om(F_I, G_I)) \xrightarrow{\sim} \operatorname{Hom}(\mathbb{C}_{M \times \{t\}}, \mathcal{H}om(F_I, K_s^!G_I)). \]

Furthermore, $\operatorname {Hom}(\mathbb {C}_{M \times \{t\}}, \mathcal {H}om(F_I, K_s^!G_I))$ is independent of $t$. The same is true if we replace $\{t\}$ by any subinterval, e.g. $[a,b]$ and $(a,b)$ of $I$.

Proof. Unwinding the definition of $K_s^!$, we have

\begin{align*} &\operatorname{Hom}(\mathbb{C}_{M \times \{t\}}, \mathcal{H}om(F_I, K_s^!G_I)) \\ &\quad=\operatorname{Hom}(\mathbb{C}_{M \times \{t\}}, \mathcal{H}om(F_I, \pi_{1*} \mathcal{H}om(K_s, \pi_2^! G_I))) \\ &\quad=\operatorname{Hom}(\mathbb{C}_{M \times \{t\}}, \pi_{1*} \mathcal{H}om(\pi_1^* F_I, \mathcal{H}om(K_s, \pi_2^! G_I))) \\ &\quad=\operatorname{Hom}(\pi_1^* \mathbb{C}_{M \times \{t\}}, \mathcal{H}om(K_s, \mathcal{H}om(\pi_1^* F_I, \pi_2^! G_I))). \end{align*}

We claim that

(12)\begin{equation} SS^\infty (\pi_1^* \mathbb{C}_{M \times \{t\}}) \cap SS^\infty \mathcal{H}om(K_s, \mathcal{H}om(\pi_1^* F_I, \pi_2^! G_I)) = \emptyset \quad \text{for all } 0< s<\epsilon. \end{equation}

The verification is straightforward, though a bit tedious, and we leave it to the reader.

From this claim and the fact that

\begin{align*} &\operatorname{Hom}(\pi_1^* \mathbb{C}_{M \times \{t\}}, \mathcal{H}om(K_s, \mathcal{H}om(\pi_1^* F_I, \pi_2^! G_I)))\\ &\quad \simeq \operatorname{Hom}(\pi_1^* \mathbb{C}_{M \times \{t\}} \otimes K_s, \mathcal{H}om(\pi_1^* F_I, \pi_2^! G_I)), \end{align*}

we may apply Lemma 2.5(2) on the shrinking closed set to get

\[ \operatorname{Hom}(\pi_1^* \mathbb{C}_{M \times \{t\}} \otimes K_s, \mathcal{H}om(\pi_1^* F_I, \pi_2^! G_I)) \simeq \operatorname{Hom}(\pi_1^* \mathbb{C}_{M \times \{t\}} \otimes K_0, \mathcal{H}om(\pi_1^* F_I, \pi_2^! G_I)) \]

for all $0< s<\epsilon$. This proves the first statement of the lemma.

The statement about independence of $t$ follows from (12) and Proposition 2.8.

The subinterval case can be proved similarly, and we omit the details.

Now we finish proving the proposition. By Lemma 3.2,

\[ \operatorname{Hom}(\mathbb{C}_{M \times (a, b )}, \mathcal{H}om(F_I, G_I)) \]

is independent of $(a,b)$, so we may shrink from $(0,1)$ to an arbitrary small neighborhood of $t$. Then we have

\begin{align*} \operatorname{Hom}(F_I, G_I) &\simeq [\pi_{I*} (\mathcal{H}om(F_I, G_I))]_t \simeq [\pi_{I*} (\mathcal{H}om(F_I, K_s^! G_I))]_t \\ &\simeq \operatorname{Hom}(\iota_t^*F_I,\iota_t^* K_s^! G_I) \simeq \operatorname{Hom}(F_t,R^s G_t) \simeq \operatorname{Hom}( F_t, G_t), \end{align*}

where $0< s<\epsilon$ and we have used a small Reeb perturbation to make $F_I, K_s^! G_I$ a non-characteristic isotopy of sheaves and then applied (6) from the proof of Proposition 2.8.

Proposition 3.3 Let $\{\Lambda ^\infty _t\}$ be a family of Legendrians in $T^\infty M$ that are uniformly displaceable with parameter $\epsilon$. For a given $t$, let $F_t \in Sh(M, \Lambda ^\infty _t)$. Suppose we have $F'_I$ and $F''_I$ in $Sh(M_I, \Lambda ^\infty _I)$ and isomorphisms

\[ f: F'_I|_t \xrightarrow{\sim} F_t, \quad g: F''_I|_t \xrightarrow{\sim} F_t. \]

Then there exists a canonical isomorphism

\[ \Phi: F'_I \to F''_I \]

such that $\Phi |_{t} = g^{-1} \circ f: F'_I|_t \to F''_I|_t$.

Proof. The proof follows from Proposition 3.1 by standard arguments.

3.2 Existence of local extension

Proposition 3.4 Let $I=[0,1]$. Let $\mathcal {L}_I$ be a strong isotopy of Legendrians in $S^*M \times T^*I$ with the slice over $t$ denoted by $\mathcal {L}_t$. Let $(U_I, X_I)$ be a strong isotopy of convex tubes for $\mathcal {L}_I$. Then for any $t \in I$ and $F_t \in Sh(M, \mathcal {L}_t)$, there exists an interval $J \supset t$ and $F_J \in Sh(M_J, \mathcal {L}_J)$ such that $F_J|_t = F_t$, where $M_J = M \times J$ and $\mathcal {L}_J = \mathcal {L}_I \cap S^*M \times T^*J$.

Proof. For any interval $J \subset I$, let $U_J = U_I \cap S^*M \times T^*J$. Then for small enough $J$ containing $t$, we have $\mathcal {L}_{t} \times T^*_J J \subset U_J$. Let $X_J$ denote the restriction of $X_I$ to $X_J$; then if we define (see Proposition 2.13 for definition of $\Pi _{U,X}$)

\[ F_J := \Pi_{(U_J, X_J)}(F_t \boxtimes \mathbb{C}_J), \]

we have $F_J|_t = F_t$ and $SS^\infty (F_J) \in \mathcal {L}_J$.

3.3 Proof of Theorem 0.5

By the local extension result (Proposition 3.4) and uniqueness of extension result, for any $t \in I=[0,1]$ and $F_t \in Sh(M, \mathcal {L}_t)$ we can extend $F_t$ to $F_I \in Sh(M_I, \mathcal {L}_I)$ such that $F_I|_t=F_t$. Hence the functor $\iota _t^*$ is fully faithful (Proposition 3.1) and essentially surjective; thus it is an equivalence.