Open Mathematics Courses

Constructible and perverse sheaves

A sheaf records local data and the maps that make those data compatible. This course asks how that description behaves at singular boundaries, under geometric transport, and in global index calculations. Its examples connect finite diagrams and small neighborhoods to cotangent geometry, duality and perverse sheaves.

Begin with Local models and finite data, then use Geometry of singular sets, Constructibility under operations, and Duality and categorical degrees to establish the common tools. From there, follow the route through degeneration and perversity, the route through kernels and microlocal coefficients, or the route through chains and index formulas. Fixed-point traces and Stein vanishing bring these tools together.

These are reading routes, not a claim that every prerequisite belongs to the same route. Follow the proof links within each lesson when a construction uses a result from elsewhere. Familiarity with sheaves of modules, chain complexes and derived functors is assumed; the programme’s derived-category lessons provide that starting point.

Choose a reading route

Local models and finite data

Start with an interval, a simplex and a small ball. These models explain what a stalk records, how restriction maps glue local data, and which finiteness hypotheses survive passage to cohomology.

Work through the interval and triangulation calculations before using a general constructibility theorem. Keep ordinary sheaves, complexes and perfect coefficients distinct.

  1. Constructible gluing on an interval
  2. Directional tests at a constructible boundary
  3. Constructible sheaves on a triangulation
  4. Small balls, central fibres and supported cohomology
  5. Perfect coefficients on compact fibres
  6. Local systems across an analytic boundary
  7. Constructible models in one cotangent direction

Geometry of singular sets

The next question is geometric: where can a sheaf change? Analytic branches, subanalytic parametrizations and limiting tangent directions make singular boundaries precise. Conormal and isotropic calculations then turn those boundaries into cotangent conditions.

Use the analytic and subanalytic foundations as references throughout the course. The literature guide helps locate complementary treatments; the lesson links supply the particular arguments used here.

  1. Weierstrass parametrization and connected regular loci
  2. Subanalytic sets and limiting tangent directions
  3. Conic subanalytic images and isotropic dimension
  4. Isotropic cotangent transport and discrete critical values
  5. Finite conormal closures and generic base directions
  6. Involutive subsets of subanalytic isotropic sets
  7. Boundary forms and Lagrangian normal cones
  8. Limiting cotangent sums and characteristic inverse images
  9. Microlocal stratifications by removing bad loci
  10. Unshared conormal directions and dimension filtrations
  11. Generic squared distance and cotangent transversality
  12. Whitney secants and the microlocal stratification condition
  13. Closed conormal unions at singular points
  14. Reading the geometry and cycle literature

Constructibility under operations

Determine when an operation preserves the chosen class of coefficients. Begin with a triangulated description, then compare it with a condition on microsupport. The complex analytic examples expose the difference between an ordinary constructible object and a derived category with all the required morphisms.

For each theorem record the coefficient ring, boundedness, properness and finiteness assumptions. A proper-on-support statement and a globally proper statement answer different questions.

  1. Constructibility from microsupport and perfect stalks
  2. Derived constructibility through common triangulations
  3. Weak constructibility under sheaf operations
  4. Perfect operations and finite microlocal coefficients
  5. Constructibility through smooth cutoffs and microlocal properness
  6. Complex conicity and analytic Lagrangian closures
  7. Analytic normal cones through complex deformation
  8. Analytic conormal covers and singular involutivity
  9. Complex microlocal stratifications and constructibility
  10. Holomorphic operations and complex Fourier symmetries
  11. Nonproper holomorphic pushforwards through cutoffs
  12. Local holomorphic pushforwards over a complex curve
  13. Complex-constructible sheaves and missing derived classes

Duality and categorical degrees

Construct the dualizing object from oriented local pieces and track the evaluation maps through direct and inverse images. Truncation and adjunction then explain how cohomological degree becomes an abelian-category structure.

Keep the actual comparison map visible. An isomorphism between two possible target objects does not by itself identify the natural transformation being studied.

  1. The dualizing complex from oriented simplices
  2. Constructible costalks and Verdier duality
  3. Duality maps for constructible inverse and direct images
  4. Natural duality for specialization and microlocal Hom
  5. Truncation triangles and abelian hearts
  6. T-exact functors and adjoints between hearts

Degeneration and perversity

Let the geometry vary through a special fibre. Nearby and vanishing cycles record the resulting change and its monodromy. Support and costalk inequalities reorganize these complexes into perverse objects, with fibre dimensions controlling their transport.

Follow both maps in each monodromy triangle and calculate the shifts in a small local model before applying a global statement.

  1. Nearby cycles and the two monodromy triangles
  2. Proper pushforwards of nearby and vanishing cycles
  3. Nearby cycles through the normal deformation
  4. Complex nearby cycles as normal and conormal sections
  5. Quadratic cycles and the holomorphic microsupport test
  6. Vanishing cycles as positive real support
  7. Perverse support, costalks and truncation triangles
  8. Perverse descent and fibre dimension bounds
  9. Complex middle perversity and exterior products

Kernels as operators

Treat a sheaf on a product as an operator between sheaf categories. Start with its support correspondence and adjoints, then test invertibility in chosen cotangent directions. Ball, incidence and Legendre kernels provide explicit models for the general constructions.

This is one of two longer routes after the local and geometric foundations. Draw the projections of each product and track which supports make their pushforwards available.

  1. Sheaf kernels and cotangent correspondences
  2. Kernels that preserve chosen cotangent directions
  3. Adjoints of localized sheaf kernels
  4. Dual kernels and an unchanged parameter
  5. Directional morphisms through a sheaf kernel
  6. When a kernel quantizes a contact transformation
  7. Real and complex ball kernels
  8. Local existence of contact kernel equivalences
  9. Projective incidence as a contact kernel
  10. The partial Legendre kernel and its adjoint shift
  11. Fourier-Sato transport of microlocal morphisms
  12. Composing hypersurface kernels with their shifts

Microlocal coefficients and composition

Once a kernel gives a local equivalence, ask what it does to a single coefficient and its degree. Pure and simple types turn composition, transverse restriction and direct or inverse image into precise statements about local modules and shifts.

Compare the orientation line and the cohomological shift separately. Tensor, Hom and composition require their own transversality hypotheses.

  1. Pure and simple sheaves from directional tests
  2. Coefficients in a quantization of the identity
  3. Normal forms and the shift of a submanifold transform
  4. How simple sheaf shifts change along a Lagrangian
  5. Tensor and Hom types at independent covectors
  6. Microlocal composition at prescribed covectors
  7. The type and shift of a transverse kernel composition
  8. Direct and inverse images of a sheaf type
  9. Transverse restrictions of microlocal coefficients

Chains and characteristic cycles

The second longer route turns local sheaf information into a geometric cycle. Build chains and their supports first; develop supported intersection and pullback next; only then assemble characteristic cycles and test their signs on half-lines and cones.

Read the support conditions together with the chain maps. Proper transport, a well-defined intersection and a numerical intersection are separate steps.

  1. Subanalytic chains and closed cycle supports
  2. Supports, products and proper images of chains
  3. The dualizing resolution by subanalytic chains
  4. Intersections of supported subanalytic cycles
  5. Continuous sections and supported cycle intersections
  6. Pulling back Lagrangian cycles through a graph
  7. Transverse pullback of normalized conormal cycles
  8. Orientations of conormal cycles and transverse intersections
  9. Lagrangian cycles and proper cotangent images
  10. Characteristic cycles from supported microlocal identities
  11. Transporting characteristic cycles through a graph
  12. Integer coefficients and additive characteristic cycles
  13. Antipodal duality and half-line characteristic cycles
  14. Lorentz cones and their vertex characteristic cycles

Euler integration and index formulas

Pass from complexes to integer-valued functions and compare their integral operations with characteristic cycles. Euler integration, convolution, vector-field indices and Morse models provide calculations in which the categorical and geometric routes meet.

Start with compact examples. Before extending a formula to a noncompact space, identify the exact support or proper-below condition that makes the index meaningful.

  1. Constructible functions and Euler integration
  2. Complex stalk–costalk Euler numbers and function duality
  3. Homogeneous functions, Fourier transformation and specialization
  4. Euler convolution and convex inverses
  5. Constructible functions and integral Lagrangian cycles
  6. Euler numbers and vector-field indices
  7. Constructible traces and local Euler indices
  8. Closed supports and evaluated proper transport
  9. Proper characteristic classes and the compact index
  10. Differential sections and proper-below Euler indices
  11. Isolated phases and local characteristic-cycle indices
  12. Pure test degrees and strong Morse inequalities
  13. Compact Morse models and full cotangent projections

Fixed points and complex vanishing

Apply the preceding constructions to a correspondence and its fixed set. Local cutoffs and tangent specialization isolate its trace contribution; expanding and shrinking directions explain localization. Holomorphic Morse exhaustions provide a final application to global cohomological vanishing.

Return to the local coefficient and orientation calculations when evaluating a contribution. The global formula retains the choices and hypotheses established there.

  1. Lefschetz traces of constructible correspondences
  2. Homotopies and local cutoffs for Lefschetz contributions
  3. Specializing Lefschetz contributions to the tangent space
  4. Expanding subspaces and hyperbolic Lefschetz cutoffs
  5. Shrinking localization and complex fixed-point traces
  6. Holomorphic Morse exhaustions on Stein manifolds
  7. Microlocal types and Stein cohomology

Working through a lesson

For each result, write down its objects, coefficient hypotheses and natural map. Test the statement on one of the finite or local examples, then work through the proof and the solved exercises. When moving to a new setting, identify which argument establishes existence, which gives finiteness, and which controls signs or degrees.

Original text CC0 1.0 unless the lesson states other terms.