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
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.
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 .
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
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 ,
Example 1 For 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 3 If 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 2 If is a diffeo, this amounts to the usual lift of to and .
Example 3 If 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
Note the sign in the contact structure. This is contactomorphic with .
Definition 4 A 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 4 Given , let is a contact correspondance.
Example 5 The graph of a contactomorphism
defines a correspondance.
More generally, a contact isotopy with Hamiltonian has a Legendrian graph
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 6 A 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 .