## Notes of Patrick Massot’s lecture

Microsupport of sheaves in symplectic topology

1. Goal of the working seminar

Use sheaves to prove a weak form of one of Arnold’s conjectures.

If ${\phi}$ is a hamiltonian isotopy in ${T^* N}$, then ${N\cap \phi_t(N)\not=\infty}$ for all ${t}$.

Proved by Hofer, and reproved by Laudenbach-Sikorav. The sheaf-theoretic proof is due to Tamarkin, Guillermou, Kashiwara-Shapira.

Sources : Claude Viterbo’s lecture notes.

A more remote goal is Stéphane Guillermou’s 2012 preprint on quantification of Lagrangian submanifolds.

2. The bundle of contact elements

2.1. Definition

${M}$ a manifold. ${\mathcal{C}M=\{}$ cooriented hyperplanes in ${TM\}}$. It carries a tautological contact structure ${\xi_H=\pi_*^{-1}(H)}$.

If ${Z\subset M}$ is a closed submanifold, let

$\displaystyle \begin{array}{rcl} \mathcal{N}Z=\{H\,;\,H\supset TZ\}. \end{array}$

If ${Z}$ has codimension 1 and is cooriented, one can define the subset ${\mathcal{N}_+ Z\subset\mathcal{N}Z}$ of positive elements. These are Legendrian submanifolds.

2.2. Lifting

If ${L\subset T^* N}$ is exact Legendrian, i.e. ${\lambda_{|L}=df}$, then ${L}$ lifts to

$\displaystyle \begin{array}{rcl} \hat{L}=\{(q,-f(p,q)),[p,1])\,;\,(q,p)\in L\}, \end{array}$

which is Legendrian in ${\mathcal{C}(M\times{\mathbb R})}$.

Exact hamiltonian isotopies of ${\mathcal{C}M}$ lift to ${\mathcal{C}(M\times{\mathbb R})}$ as well.

The symplectization of ${(V,\xi)}$ is

$\displaystyle \begin{array}{rcl} S(V,\xi)=\{(q,p)\in T^*V\,;\,\mathrm{ker}(p)=\xi_q\}. \end{array}$

Contact isotopies correspond to ${{\mathbb R}_+}$-invariant functions on ${S(V,\xi)}$ in the following manner : ${\phi_t}$ correspond to ${H_t(q,p)=p\circ(T_q\phi_t)^{-1}}$.

Special case of contact elements: ${S(\mathcal{C}(M),\xi_{can})=T^*M\setminus 0_M}$.

The lift ${\hat{\phi}_t}$ of ${\phi_t}$ arises from the Hamiltonian

$\displaystyle \begin{array}{rcl} \hat{H}_t((n,s,p_n,p_s))=p_s H_t(n,\frac{p_n}{p_s}). \end{array}$

If ${\phi_t}$ has compact support, then ${\hat{\phi}_t}$ exists on the whole of ${\mathcal{C}(N\times{\mathbb R})}$.

2.3. Walls

If ${\psi:M\rightarrow{\mathbb R}}$ has no critical point, then it defines a wall in ${\mathcal{C}M}$,

$\displaystyle \begin{array}{rcl} W_\psi =\{(q,\mathrm{ker}(d_q\psi)\}\subset\mathcal{C}M, \end{array}$

which is foliated by ${\mathcal{N}\psi^{-1}(*)}$.

2.4. Contact version of the disjunction conjecture

Let ${(\phi_t)}$ be a hamiltonian isotopy with compact support. Then intersection points ${\phi_t(0_N)\cap 0_N}$ correspond to intersection points of ${\phi_t(\mathcal{L})\cap W_\psi}$, for ${\psi(n,s)=s}$.

So from now on we shall work in the contact setting only.

3. Microsupport and intersections

3.1. Strategy

There is a category whose objects generalize submanifolds of ${M}$ and local systems: this is the derived category of sheaves on ${M}$. A submanifold ${Z\subset M}$ corresponds to the constant sheaf along ${Z}$, ${k_Z}$.

If ${U}$ is a codimension 0 submanifold with smooth cooriented boundary, there is also a corresponding ${k_U}$.

Any object in this category has a support ${supp(\mathcal{F})\subset M}$. It also has a microsupport ${\mathcal{N}\mathcal{F}}$ which is a closed subset of ${\mathcal{C}M}$.

To any subset ${A\subset M}$, there is an associated cohomological object ${R\Gamma(A,\mathcal{F})}$. For instance, for the constant sheaf, ${R\Gamma(A,k_M)}$ contains the same amount of information as ${H^*(A,k)}$ when ${k}$ is a field.

Proposition 1 (Morse Lemma) Let ${\psi:M\rightarrow{\mathbb R}}$ be proper on the support of ${\psi}$. Fix ${a. Assume that for all ${x}$ such that ${a\leq \psi(x),

• either ${d\psi(x)=0}$ and ${x}$ is not in the support of ${\mathcal{F}}$,
• or ${[d\phi(x)]\notin\mathcal{NF}}$,

then

$\displaystyle \begin{array}{rcl} R\Gamma(\{\psi

Example 1 For the constant sheaf, ${\mathcal{NF}}$ is empty, this is nothing but the usual Morse Lemma.

3.2. Main Quantization Theorem

Theorem 2 (Guillermou-Kashiwara-Shapira) Suppose ${\mathcal{F}_0}$ has compact support in ${M}$. and ${\Phi}$ is a contact isotopy in ${\mathcal{C}M}$. Then there exists a family ${(\mathcal{F}_t)}$ in the category such that

$\displaystyle \begin{array}{rcl} \Phi_t(\mathcal{NF}_0)=\mathcal{NF}_t, \quad\textrm{and}\quad R\Gamma(M,\mathcal{F}_t)\simeq R\Gamma(M,\mathcal{F}_0). \end{array}$

Corollary 3 If ${\psi:M\rightarrow{\mathbb R}}$ has no critical points, ${\mathcal{F}_0}$ has compact support and ${R\Gamma(M,\mathcal{F}_0)\not=0}$, then for all ${t}$,

$\displaystyle \begin{array}{rcl} \phi_t(\mathcal{NF}_0)\cap W_\psi\not=\emptyset. \end{array}$

This clearly implies the form of Arnold’s conjecture we are aiming at. Indeed, to the 0-section ${0_N}$, there corresponds ${\mathcal{N}k_U}$ where ${U=\{(n,s)\,;\,s\leq 0\}}$. Use function ${\psi(n,s)=s}$. Then ${R\Gamma(M,k_U)\simeq H^*(N,{\mathbb R})}$.

3.3. Proof of corollary

From Theorem 1 and Morse Lemma.

The Theorem provides us with the family ${\mathcal{F}_t}$ of objects (quantification of the transported Lagrangians). By contradiction, assume that for some ${t}$, ${\phi_t(\mathcal{NF}_0)=\mathcal{NF}_t}$ does not intersect the wall ${W_\psi}$. Then Morse Lemma applies to ${\mathcal{F}_t}$ with ${a}$ negative enough in order that ${\{\psi does not intersect the support of ${\mathcal{F}_t}$, and ${b}$ large enough so that ${\{\psi. Then cohomology ${R\Gamma(M,\mathcal{F}_t)\simeq R\Gamma(M,\mathcal{F}_0)}$. This contradicts the fact that ${R\Gamma(\{\psi.

4. Quantizing isotopies

The above Theorem 1 follows from a more abstract theorem, which quantizes isotopies. The quantization can be applied to ${\mathcal{F}_0}$ to get the family ${\mathcal{F}_t}$. Next goal is to quantize ${\Phi}$, getting ${\mathcal{K}_\phi}$, such that

$\displaystyle \begin{array}{rcl} \mathcal{K}_\phi(\mathcal{F}_0)=\mathcal{F}_t. \end{array}$

Before we get into thatn we need more geometry.

4.1. Functoriality in contact tautology

Let ${f:M\rightarrow N}$ be an arbitrary (smooth) map between manifolds. It pushes forward arbitrary subsets ${L\subset\mathcal{C}M}$ in the following way,

$\displaystyle \begin{array}{rcl} f_*(L)=\{(m,[p_m])\in\mathcal{C}M\,;\,\exists (n,[p_n])\in L,\,m=f(n), \,p_m\circ T_n f=p_n\}. \end{array}$

Dually, if ${L'\subset \mathcal{C}M}$, get ${f^*L'\subset \mathcal{C}N}$.

Example 2 If ${f}$ is a diffeo, this amounts to the usual lift of ${f}$ to ${\mathcal{C}M}$ and ${\mathcal{C}N}$.

Example 3 If ${f:E\rightarrow M}$ is a submersion, ${W\subset E}$ is a cooriented hypersurface, in generic situations, ${\mathcal{L}=f_*(\mathcal{N}_+ W)}$ is a Legendrian submanifold in ${\mathcal{C}M}$, and ${W}$ is called a generating hypersurface for ${\mathcal{L}}$.

This covers much more submanifolds in ${\mathcal{C}M}$ than you might expect, see forthcoming talk by Ferrand.

4.2. Contact correspondances

Let

$\displaystyle \begin{array}{rcl} \mathcal{C}(N,M)=(\mathcal{C}(N\times M),\xi=\mathrm{ker}(-\lambda_N+\lambda_M)). \end{array}$

Note the sign in the contact structure. This is contactomorphic with ${\mathcal{C}(N\times M)}$.

Definition 4 A contact correspondance between ${\mathcal{C}(N)}$ and ${\mathcal{C}(M)}$ is a Legendrian submanifold in ${\mathcal{C}(N,M)}$.

Switching ${M}$ and ${N}$ produces a contactomorphism which map a correspondance to the dual correspondance, by definition.

Example 4 Given ${f:N\rightarrow M}$, let ${f_*=\{(n,m,[p_n,p_m])\,;\,m=f(n),\,p_m\circ T_n f=p_m\}}$ is a contact correspondance.

4.3. Graphs

Example 5 The graph of a contactomorphism ${\phi:M\rightarrow M}$

$\displaystyle \begin{array}{rcl} \Gamma_\phi=\{(x,y,[px,py]),(y,[p_y])=\phi(x,[p_x])\}. \end{array}$

defines a correspondance.

More generally, a contact isotopy ${\Phi}$ with Hamiltonian ${H_t}$ has a Legendrian graph

$\displaystyle \begin{array}{rcl} \Gamma_\Phi \subset\mathcal{C}(M,M\times I), \end{array}$

defined by

$\displaystyle \begin{array}{rcl} \Gamma_\Phi =\{(q,q',t,[p_q,p_{q'},p_t])\,;\,\phi_t(q,[p_q])=(q',[p_{q'}]),\,p_t=H_t(q,p_t)\}. \end{array}$

By quantizing a Hamiltonian isotopy ${\Phi=(\phi_t)_{t\in I}}$, we mean an object in our category corresponding to ${\Gamma_\Phi}$.

4.4. More abstract quantization

Theorem 5 (Guillermou-Kashiwara-Shapira) For any contact isotopy ${\Phi}$ in ${\mathcal{C}M}$, the is an object ${\mathcal{K}_\Phi}$ of our category on ${M\times M\times I}$ such that

$\displaystyle \begin{array}{rcl} (\mathcal{NK}_\Phi)^a=\Gamma_\Phi. \end{array}$

4.5. Microsupport functoriality

We need some more notation in order to explain how Theorem 2 implies Theorem 1.

A map ${f:N\rightarrow M}$ induces a functor ${RF_! : Db N\rightarrow D^b M}$, related to the cohomology of fibers. If ${f}$ is proper on the support of ${\mathcal{F}}$, then

$\displaystyle \begin{array}{rcl} \mathcal{N}(Rf_! \mathcal{F})\subset f_*(\mathcal{NF}). \end{array}$

Definition 6 A submanifold ${L\subset \mathcal{C}M}$ is non characteristic for a map ${f:N\rightarrow M}$ if

$\displaystyle \begin{array}{rcl} f_*(pr_M^*(L))=\emptyset \subset \mathcal{C}(N,M). \end{array}$

There is also a functor ${f^{-1}}$ in the reverse direction. If ${\mathcal{NF}}$ is non characteristic for ${f}$, then ${\mathcal{N}(f^{-1}(\mathcal{F}))\subset f^{*}(\mathcal{NF})}$.

4.6. Theorem 2 implies Theorem 1

${\mathcal{K}_{\Phi}}$ lives on ${M\times M\times I}$. It can be used to push forward ${\mathcal{F}_0}$, the resulting object ${\mathcal{F}=\mathcal{K}_\Phi(\mathcal{F}_0)}$ is defined on ${M\times I}$. Freezing at ${M\times\{t\}}$ yields ${\mathcal{F}_t =j_t^{-1}\mathcal{F}}$. Since ${\mathcal{NF}}$ is non characteristic for ${j_t}$,

$\displaystyle \begin{array}{rcl} \mathcal{NF}_t\subset j_t^*(\mathcal{NF})\subset \Gamma_{\phi_t}. \end{array}$

It fact, equality holds but we shall not need it. Cohomology does not change because ${R_{\pi_{!}}\mathcal{F}}$ is locally constant on ${I}$. This is related to the fact that its microsupport ${\mathcal{N}(R_{\pi_{!}}\mathcal{F})}$ is empty. Indeed, this is contained in the projection of the microsupport of ${\mathcal{F}=\mathcal{K}_\Phi(\mathcal{F}_0)}$. ${\mathcal{NF}}$ is contained in the composition of the correspondance ${\Gamma_\Phi}$ with ${\mathcal{NF}_0}$. This is empty because ${\mathcal{F}}$ is non characteristic for injections ${j_t}$.