Sources, calculations and prerequisite proofs
The six readings study one question from three directions: what survives near a zero fibre, how it changes under a map, and how it detects a cotangent direction. Their proofs use a fixed coefficient complex and explicit comparison maps. The distinction between a calculation performed here and a prerequisite theorem matters, especially for infinite coefficient modules.
Start with calculations that fix the conventions
Begin with the finite-support sequence calculation in the two monodromy triangles. Compute ramification before using a geometric comparison: it checks the diagonal unit and the two composites (1-M). The product–stalk example explains why a countable-cover argument needs a common system of shrinking neighborhoods.
Next, proper pushforward follows one closed support carrier through internal Hom and base change. Its graph application is the passage from a critical function to a regular ambient coordinate. Normal deformation instead compares the actual covers through the logarithmic lift. Its closed-ray example records what weak real constructibility permits before complex constructibility is imposed.
The normal and conormal section argument adds complex scaling, an endpoint-sensitive polar calculation and a specified slit. Quadratic tests then compute a sphere, its reduced cochains and its antipodal action before using a microlocal coefficient model. Finally, positive real support compares the same cycle convention with a closed halfspace using a normalized branch. Its ramification and closed-ray examples distinguish the roles of a branch, the cycle shift and complex constructibility.
What the checked human sources supply
- David B. Massey, Notes on Perverse Sheaves and Vanishing Cycles, arXiv:math/9908107v13, 17 August 2025: §3 supplies traditional and coefficient-complex definitions, the corrected punctured trace, cycle triangles, the proper-map statement and the positive-real-support comparison. The historical credit to Kashiwara and Schapira is retained through Massey’s account. His constructible setting does not by itself establish the weak-coefficient or proper-on-closed-support extensions in these lessons. The exact author TeX was checked against the official v13 source archive. The paper’s stated terms are CC BY 4.0; no passage of its expression is reproduced here.
- Ren Fernandes, Kazuki Kudomi and Kiyoshi Takeuchi, Characteristic cycles of real and complex constructible sheaves, revisited, arXiv:2603.14821v2, 10 July 2026: §2.4 defines specialization and microlocalization; (4.54) relates positive deformation to real nearby cycles. The proof of Theorem 5.5 provides a useful stratum-dimension calibration. Equations (5.42)–(5.43) state the regular-fibre conormal comparison by reference to an earlier work. These passages do not prove our full weak-coefficient comparisons, generic coefficient-model theorem or uniform holomorphic criterion. The exact v2 author TeX was checked; no source text or figure is reproduced.
- Masaki Kashiwara, Index theorem for constructible sheaves, Astérisque 130 (1985), pp. 193–209: Lemma 5.2 on printed p. 201 states the real quadratic local-support degree for vector-space coefficients. The displayed page was checked. Our arbitrary-module calculation uses its own finite free sphere-cochain model; the lemma is not cited as a proof of that extension. No page or source expression is included in the download.
The proof obligations behind the comparisons
These readings contain the local comparison arguments and forty-one solved exercises. They are not a self-contained construction of the sheaf and microlocal foundations. In particular, the following inputs retain their full original scope:
| Application | Required programme input | What remains separate |
|---|---|---|
| Coefficient construction and pushforward | Ordinary and proper-support adjunction, tensor–Hom, proper-support base change and finite-dimensional cohomological bounds | The underlying resolution, derived-category and topology foundations; a bounded ordinary adjunction argument is included in the pushforward lesson |
| Countable-cover deformation | Whole-complex small-ball stabilization, weak inverse-image and internal-Hom properties, smooth base change and ordinary/punctured conic recovery | Uniform control of the shrinking neighborhoods; a formal interchange of a product and stalk colimit does not prove it |
| Normal and conormal sections | SH02-CHE-001, SH02-CON-CYLINDER, and
SH02-FS-SECTIONS (FS13) |
The analytic normal-cone estimate, full-complex cylinder descent and Fourier section theorem at the exact weak-coefficient scope |
| Uniform holomorphic detection | SH02-LFI-SUPPORTED (LFI9–LFI10),
SH02-MC-LOCAL (MC.2), and
SH02-MO-MICROLOCAL-SUPPORT (MO15), with the earlier
complex-microsupport theorem |
Generic coefficient objects, arbitrary denominator cones, the quotient/null criterion and singular analytic geometry |
| Positive-real-support comparison | The local complex-curve theorem, cylinder descent and support localization | Their exact bounded weak-coefficient versions; source statements with finite or field coefficients do not fill the gap |
| Covered quadratic computation | Sheaf/singular cochain comparison, constant-coefficient homotopy invariance and finite sphere cochains | The full topological providers, distinct from the explicit retraction and degree calculation |
Some of these prerequisite chains are still being reconstructed. The argument at each use retains its hypotheses and identifies the required theorem; inclusion in this selection does not certify that every transitive input is complete. No claim of full course completion follows from the checked source passages.
Reuse
The independently written programme prose, calculations and solutions are dedicated to the public domain under CC0 1.0 Universal. Cited human works keep their own terms. Source access, mathematical proof and permission to reproduce source expression are separate questions.