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<b}. Assume that for all {x} such that {a\leq \psi(x)<b},

  • 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<a\},\mathcal{F})\simeq R\Gamma(\{\psi<b\},\mathcal{F}). \end{array}

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<a\}} does not intersect the support of {\mathcal{F}_t}, and {b} large enough so that {\{\psi<b\}=M}. Then cohomology {R\Gamma(M,\mathcal{F}_t)\simeq R\Gamma(M,\mathcal{F}_0)}. This contradicts the fact that {R\Gamma(\{\psi<a\},\mathcal{F}_t)=0}.

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}.

About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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