** 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 is a hamiltonian isotopy in , then for all *.

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 **

a manifold. cooriented hyperplanes in . It carries a tautological contact structure .

If is a closed submanifold, let

If has codimension 1 and is cooriented, one can define the subset of positive elements. These are Legendrian submanifolds.

** 2.2. Lifting **

If is exact Legendrian, i.e. , then lifts to

which is Legendrian in .

Exact hamiltonian isotopies of lift to as well.

The symplectization of is

Contact isotopies correspond to -invariant functions on in the following manner : correspond to .

Special case of contact elements: .

The lift of arises from the Hamiltonian

If has compact support, then exists on the whole of .

** 2.3. Walls **

If has no critical point, then it defines a wall in ,

which is foliated by .

** 2.4. Contact version of the disjunction conjecture **

Let be a hamiltonian isotopy with compact support. Then intersection points correspond to intersection points of , for .

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 and local systems: this is the derived category of sheaves on . A submanifold corresponds to the constant sheaf along , .

If is a codimension 0 submanifold with smooth cooriented boundary, there is also a corresponding .

Any object in this category has a support . It also has a microsupport which is a closed subset of .

To any subset , there is an associated cohomological object . For instance, for the constant sheaf, contains the same amount of information as when is a field.

Proposition 1 (Morse Lemma)Let be proper on the support of . Fix . Assume that for all such that ,

- either and is not in the support of ,
- or ,
then

Example 1For the constant sheaf, is empty, this is nothing but the usual Morse Lemma.

** 3.2. Main Quantization Theorem **

Theorem 2 (Guillermou-Kashiwara-Shapira)Suppose has compact support in . and is a contact isotopy in . Then there exists a family in the category such that

Corollary 3If has no critical points, has compact support and , then for all ,

This clearly implies the form of Arnold’s conjecture we are aiming at. Indeed, to the 0-section , there corresponds where . Use function . Then .

** 3.3. Proof of corollary **

From Theorem 1 and Morse Lemma.

The Theorem provides us with the family of objects (quantification of the transported Lagrangians). By contradiction, assume that for some , does not intersect the wall . Then Morse Lemma applies to with negative enough in order that does not intersect the support of , and large enough so that . Then cohomology . This contradicts the fact that .

**4. Quantizing isotopies **

The above Theorem 1 follows from a more abstract theorem, which quantizes isotopies. The quantization can be applied to to get the family . Next goal is to quantize , getting , such that

Before we get into thatn we need more geometry.

** 4.1. Functoriality in contact tautology **

Let be an arbitrary (smooth) map between manifolds. It pushes forward arbitrary subsets in the following way,

Dually, if , get .

Example 2If is a diffeo, this amounts to the usual lift of to and .

Example 3If is a submersion, is a cooriented hypersurface, in generic situations, is a Legendrian submanifold in , and is called a generating hypersurface for .

This covers much more submanifolds in than you might expect, see forthcoming talk by Ferrand.

** 4.2. Contact correspondances **

Let

Note the sign in the contact structure. This is contactomorphic with .

Definition 4A contact correspondance between and is a Legendrian submanifold in .

Switching and produces a contactomorphism which map a correspondance to the dual correspondance, by definition.

Example 4Given , let is a contact correspondance.

** 4.3. Graphs **

Example 5The graph of a contactomorphism

defines a correspondance.

More generally, a contact isotopy with Hamiltonian has a Legendrian graph

defined by

By quantizing a Hamiltonian isotopy , we mean an object in our category corresponding to .

** 4.4. More abstract quantization **

Theorem 5 (Guillermou-Kashiwara-Shapira)For any contact isotopy in , the is an object of our category on such that

** 4.5. Microsupport functoriality **

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

A map induces a functor , related to the cohomology of fibers. If is proper on the support of , then

Definition 6A submanifold is non characteristic for a map if

There is also a functor in the reverse direction. If is non characteristic for , then .

** 4.6. Theorem 2 implies Theorem 1 **

lives on . It can be used to push forward , the resulting object is defined on . Freezing at yields . Since is non characteristic for ,

It fact, equality holds but we shall not need it. Cohomology does not change because is locally constant on . This is related to the fact that its microsupport is empty. Indeed, this is contained in the projection of the microsupport of . is contained in the composition of the correspondance with . This is empty because is non characteristic for injections .