Microlocal sheaf theory asks not only where a sheaf changes, but in which cotangent directions that change can be detected. It turns local propagation of information into geometry inside a cotangent bundle, where singular support behaves like a conic Lagrangian shadow of the sheaf. This makes sheaves, symplectic geometry and representation theory meet in one language.
A constructible sheaf can be perfectly constant on large regions and still carry nontrivial information at boundaries, strata and singularities. Ordinary support records where sections live. Singular support, or microsupport, records directions in which local sections fail to propagate. That extra directional information is what makes the theory powerful.
This article connects the BTT routes through Derived Categories, The Orbit Method, Springer Correspondence, and Geometric Satake.
Support · Singular support · Worked boundary example · Lagrangian geometry · Functorial rules · Representation theory · Practice
Ordinary support is not enough
Let F be a sheaf on a manifold M. The support Supp(F) is the closure of points where the stalk is nonzero. This tells us where the sheaf exists, but not how it changes.
For example, the constant sheaf on a closed half-line [0,∞) is supported on that half-line. The only geometric singularity occurs at the boundary point 0, but ordinary support does not record which directions toward or away from the boundary are responsible for the failure of propagation.
Microlocal theory refines this by moving from M into its cotangent bundle T*M.
Cotangent vectors as directional tests
A cotangent vector ξ∈T_x^*M can be represented locally as the differential dφ(x) of a function φ. The level sets of φ define hypersurfaces, and ξ records the infinitesimal normal direction.
Microlocal sheaf theory asks whether sections of F can be propagated across such a hypersurface near x. If propagation fails in the ξ direction, then (x,ξ) belongs to the microsupport SS(F).
This idea converts a sheaf-theoretic obstruction into a subset of cotangent space.
Singular support / microsupport
The singular support SS(F) is a closed conic subset of T*M. “Conic” means that if (x,ξ) belongs to SS(F), then so does (x,tξ) for every positive scalar t.
At points where F is locally constant, no nonzero cotangent direction is singular. Thus SS(F) contains only the zero section over such regions.
At a stratification boundary, conormal directions to the strata appear.
For a sheaf constructible with respect to a Whitney stratification, SS(F) is contained in the union of conormal bundles to the strata.
Worked example: the constant sheaf on a half-line
Let M=R and F=k_{[0,∞)}, the constant sheaf on the closed half-line extended by zero outside.
Away from 0, the sheaf is locally constant on (0,∞) and zero on (−∞,0). The only nontrivial microlocal behaviour occurs at the boundary.
At x=0, the cotangent fiber is one-dimensional. Depending on open/closed convention, one half-ray of nonzero covectors detects the failure of propagation across the boundary.
The important lesson is structural: ordinary support sees one boundary point; singular support sees a boundary point plus a preferred cotangent direction.
Conormal bundles
If S⊂M is a smooth submanifold, the conormal bundle is
T^*_S M={(x,ξ):x∈S, ξ(v)=0 for every v∈T_xS}.
These are cotangent directions normal to the stratum.
Conormal bundles are Lagrangian submanifolds of T*M. This is the first key bridge from sheaf theory to symplectic geometry.
Why singular support is Lagrangian-like
The cotangent bundle T*M carries its canonical symplectic form. A smooth Lagrangian submanifold has half the dimension of T*M and vanishing symplectic form when restricted to it.
For constructible sheaves, singular support is involutive; in the complex algebraic setting its irreducible components are often conic Lagrangian subsets.
This deep theorem means that directional failure of sheaf propagation is constrained by Hamiltonian geometry. Singularities do not occupy arbitrary cotangent subsets.
Microlocal stalks
Near a smooth point of SS(F), one can define a microlocal stalk or Morse group measuring the jump of sections across a hypersurface normal to ξ.
Ordinary stalks ask “what does the sheaf contain at x?” Microlocal stalks ask “what new information appears when crossing x in the ξ direction?”
This makes microlocal stalks directional analogues of local cohomology groups.
Morse theory enters naturally
Choose a function φ whose differential at x represents ξ. Compare sections over sublevel sets φ<c and φ<c+ε.
If cohomology changes as the level crosses c, the sheaf detects the covector dφ(x).
Thus microlocal sheaf theory can be understood as a sheaf-theoretic refinement of Morse theory: critical behaviour is recorded in cotangent directions rather than only at critical points.
How microsupport behaves under functors
Derived sheaf operations have microlocal estimates. Pullback, proper pushforward, tensor product and RHom transform singular support according to geometric correspondences in cotangent bundles.
For a smooth map f:X→Y, cotangent vectors pull back along df^*. Singular support of f^{-1}F is controlled by the pullback of SS(F), provided non-characteristic hypotheses are satisfied.
For proper pushforward, singular directions are projected through the cotangent correspondence associated with f.
These estimates make singular support a bookkeeping device for whether derived functors can create or remove singularities.
Non-characteristic maps
A map is non-characteristic for F when its conormal directions avoid SS(F) except at the zero section.
This is the sheaf-theoretic analogue of a PDE surface being non-characteristic.
Under non-characteristic conditions, restriction behaves cleanly and no unexpected singular propagation occurs.
The parallel with differential equations is not accidental: microlocal sheaf theory was developed alongside microlocal analysis and D-module theory.
Characteristic varieties and D-modules
A coherent D-module has a characteristic variety inside T*X obtained from the associated graded module under the order filtration.
For holonomic D-modules, the characteristic variety is Lagrangian. Under the Riemann–Hilbert correspondence, regular holonomic D-modules correspond to constructible sheaf complexes, and their characteristic geometry matches singular-support geometry.
This article therefore complements rather than duplicates BTT’s Beilinson–Bernstein route: the existing article owns localization, while this one owns directional sheaf singularity and cotangent propagation.
Perverse sheaves
Perverse sheaves are special complexes of sheaves satisfying cohomological support and cosupport conditions relative to dimension.
They behave like an abelian heart inside a derived category while retaining geometric singularity information.
Springer theory, geometric Satake and intersection cohomology use perverse sheaves because they package singular geometry in a representation-theoretically rigid way.
Microsupport adds another layer: it says which conormal directions those perverse sheaves occupy.
Representation theory from singular support
Representation categories often arise from sheaves on stratified spaces. Their simple objects are indexed by strata, orbits or local systems. Singular support places those objects inside cotangent geometry.
Characteristic cycles attach integer multiplicities to irreducible Lagrangian components of singular support. These multiplicities can encode composition-theoretic or representation-theoretic information.
For flag varieties, characteristic cycles of localization objects connect representation theory of Lie algebras with conormal geometry to Schubert strata.
Thus multiplicities in algebra can become multiplicities of geometric Lagrangian components.
Sheaf quantization of Lagrangians
A striking modern principle is that suitable Lagrangian subsets of cotangent bundles can be “quantized” by sheaves whose singular support lies in those Lagrangians.
This converts symplectic geometry into categories of sheaves. Under favorable hypotheses, Hamiltonian isotopies induce equivalences of associated sheaf categories.
This is one route toward the Nadler–Zaslow correspondence and homological mirror symmetry, where constructible sheaves and Fukaya-type categories become equivalent descriptions.
Legendrian boundaries
Because SS(F) is conic, one may quotient its nonzero directions by positive scaling and obtain a subset in the cosphere bundle S^*M.
Lagrangian cones project to Legendrian subsets there. Microlocal sheaves with prescribed singular support can therefore define invariants of Legendrian knots and links.
This is an example of representation mathematics reaching into contact topology through categorical data rather than matrices alone.
A verification workflow
- 1. Fix the manifold and sheaf category.
- 2. Identify a constructible stratification.
- 3. Separate ordinary support from singular support.
- 4. Compute conormal directions to relevant strata.
- 5. Check conicity in cotangent fibers.
- 6. Use microlocal Morse tests for directional propagation.
- 7. State non-characteristic hypotheses before restriction.
- 8. Distinguish sheaf microsupport from D-module characteristic variety.
- 9. Track Lagrangian components and characteristic-cycle multiplicities.
- 10. Keep derived-category shifts and perversity conventions explicit.
Common mistakes
- Confusing ordinary support with singular support.
- Forgetting that microsupport lives in T*M, not M.
- Ignoring the zero section over locally constant regions.
- Treating every cotangent subset as possible singular support.
- Applying pullback formulas without non-characteristic conditions.
- Calling perverse sheaves ordinary sheaves in degree zero.
- Equating D-modules and sheaves without the hypotheses of Riemann–Hilbert.
- Dropping orientation/sign conventions in half-space examples.
Practice questions
1. What does ordinary support record? 2. Where does singular support live? 3. Why is it conic? 4. What is a conormal bundle?
5. Why does a constructible sheaf have singular support in conormals to strata? 6. What does a microlocal stalk measure? 7. What is a non-characteristic map? 8. How does Morse theory enter?
9. What geometric object corresponds to a holonomic D-module? 10. Why are characteristic cycles useful? 11. What happens after quotienting conic directions by positive scaling? 12. How can sheaves quantize Lagrangians?
Worked answers
1. The closure of points where stalks are nonzero.
2. In the cotangent bundle T*M.
3. Direction matters but positive rescaling of a covector does not change the local propagation test.
4. Covectors vanishing on tangent directions to a submanifold.
5. Local variation is confined to directions normal to the chosen strata.
6. The cohomological jump when crossing a hypersurface in a specified cotangent direction.
7. A map whose conormal directions avoid SS(F) away from the zero section.
8. Sublevel-set cohomology changes detect singular cotangent directions.
9. Its characteristic variety, which is Lagrangian in the holonomic case.
10. They record multiplicities of Lagrangian components and can reflect algebraic composition data.
11. One obtains Legendrian directions in the cosphere bundle.
12. By constructing sheaves whose singular support is constrained to the chosen Lagrangian.
Sources and further study
Kashiwara and Schapira’s Sheaves on Manifolds is the foundational reference for microsupport. Nadler and Zaslow connect constructible sheaves with Fukaya categories, while modern work on microlocal sheaves applies these ideas to symplectic and contact topology. For representation theory, characteristic-cycle methods link sheaves on flag varieties with Lie-theoretic categories.
Representation Mathematics — Batch 12
Move from cotangent singularity to reflection-deformed differential operators in Rational Cherednik Algebras. Generalize the deformation picture in Symplectic Reflection Algebras. Then move to Cluster Categories and 2-Calabi–Yau Representation Theory. Return to the BTT Mathematics Learning Hub.