MICROSUPPORT SS(F) // KASHIWARA-SCHAPIRA // NON-CHARACTERISTIC DEFORMATION // COTANGENT PROPAGATION
The microsupport SS(F) of a sheaf F on a manifold X is a closed conic subset of the cotangent bundle T*X that measures where F fails to propagate. A covector (x, xi) lies in SS(F) if and only if sections of F cannot be extended past x in the direction xi.
Kashiwara-Schapira proved that SS(F) is always co-isotropic (involutive) — its dimension is at least dim X. This is the microlocal analog of the involutivity of characteristic varieties of D-modules.
The non-characteristic deformation lemma is the engine of microlocal sheaf theory. If a family of open sets {U_t} deforms along a direction non-characteristic to SS(F), then sections of F propagate along the deformation.
This reduces global computations to local ones — compute stalks, then propagate. The lemma is the microlocal incarnation of the principle that information in a sheaf flows perpendicular to its microsupport.
Classical Morse theory studies how sublevel sets X_{f<=c} change as c passes through critical values. Microlocal Morse theory replaces critical points with the sheaf-theoretic notion: the change in R Gamma(X_{f<=c}, F) occurs exactly when the level set {f=c} becomes characteristic with respect to SS(F).
The Euler obstruction and the local Euler characteristic of F at a critical point are computed via the microlocal index formula. This unifies classical Morse theory, stratified Morse theory, and constructible sheaf theory into one framework.
Tamarkin and Guillermou showed that microlocal sheaf theory provides a sheaf-theoretic approach to symplectic topology. A Lagrangian submanifold L in T*X can be "quantized" to a sheaf F with SS(F) = L. This gives sheaf-theoretic proofs of Arnold conjecture, non-displaceability, and capacities.
The microlocal stalk of F at (x, xi) in T*X is the colimit of sections along half-spaces perpendicular to xi. For a constructible sheaf on a stratified space, the microlocal stalk computes the local contribution of a stratum to cohomology — this is the "microscope" that sees individual strata.