** Singular support **

Today, the goal is to define terms arising in the statement of the Morse Lemma for sheaf cohomology.

**1. Definitions, examples **

** 1.1. Propagation and micro-support **

Given a sheaf , for every open set , is defined via injective resolutions. The stalk at point is the set of germs of sections of

Definition 1Let be a manifold, , . Say that propagates at in the direction of if for all functions such that and , for all , the natural map

is an isomorphism.

Patrick Massot prefers an alternative definition. The above definition includes the case , i.e. the support of . The following avoids this.

*Say that propagates at through , a co-oriented hyperplane in , for all domains touching and contained on the preferred side of , for all , the map
*

* is an isomorphism.*

For , it means that one-sided germs of sections uniquely extend to ordinary germs.

Definition 2The micro-support (often called singular support) of is the closure in of the set of where does not propagate. Notation: .

- It is a cone.
- The projection of is contained in the support of .
- The intersection of with the zero section equals the support of (indeed, then one can take , then for sufficiently small ; this implies that ).

** 1.2. Examples **

Example 1Let be the constant sheaf. Then equals the zero section.

Indeed, one can choose a basis of neighborhoods such that and is contractible. These open sets have equal cohomologies, so the isomorphism takes place.

Example 2Let be the constant sheaf on a closed, smooth domain . Then on the conormal to at equals the conormal of the boundary.

By definition, consists of locally constant functions on . Again, if is not conormal to at , one can choose a basis of neighborhoods such that and is contractible, the isomorphism takes place, propagation holds. Otherwise, take for a local correctly oriented equation of the boundary, and . Then, for all neighborhoods , empty, so cohomology vanishes, isomorphism fails for , no propagation.

For the next example, the following lemma will be useful.

Lemma 3For all short exact sequences of sheavesfor all permutations of indices ,

Proof uses the long exact sequence in cohomology and Five Lemma.

Example 3Let be the constant sheaf on an open domain with smooth boundary. Then .

Remember that sections of on are locally constant functions on that vanish near . By construction, there is a short exact sequence

The Lemma implies that is contained in and contains the union of and .

Example 4Let be the constant sheaf on a closed submanifold. Then equals the conormal space to .

Locally on can pick a submanifold such that . Use the exact sequence

Question: are there easy examples where non propagation is seen in higher cohomology ? Consider non smooth closed sets.

The idea of conormal sets extends to mildly singular sets, like squares.

** 1.3. Micro-support for objects of the derived category **

**Notation**. For a closed set, define a sheaf by

For open,

Lemma 4With this notation, the isomorphismholds if and only if for all ,

Indeed, there is a short exact sequence

Note that can be denoted by . This variant of the definition of propagation opens the way to generalization to the derived category of sheaves.

Definition 5Let be an object of the bounded derived category of sheaves on , . The micro-support is the closure of all such that there exists a smooth function with , such that

Alexandru Oancea: what does exactly mean ?

Example 5Let be a complex manifold. Let denote the sheaf of holomorphic functions on . Let be a differential operator with holomorphic coefficients and let be its principal symbol (a function on ). Consider the following complex of sheavesThen

This is not a easy fact. The Cauchy-Kowalewskaya theorem is used. It motivated initially the concept of micro-support.

** 1.4. Direct image **

Let , be symplectic manifolds. Let be a Lagrangian submanifold, then, for every subset , is the projection on of the intersection of with diagonal in .

Proposition 6Let be a map which is proper on the support of . Then

this is associated with the graph of which is a Lagrangian submanifold of .

Note that

Proof relies on , and a passage to the derived category which uses spectral sequences. Assume that propagates at , , for all . Let satisfy , . One needs to show that

Take . By assumption, for all . One first shows that

and, second, that

One uses the following fact.

Theorem 7Let , be left-exact functors between categories having enough injectives. Assume that send injectives to -acyclic objects. Then

C’est la preuve de ce théorème qui utilise des suites spectrales. Par exemple, le théorème implique la suite spectrale de Leray pour les fibrations.