The convolution algebra of constructible sheaves

Mehdi Benchoufi

arxiv: 2604.23872 · v2 · submitted 2026-04-26 · 🧮 math.AG

The convolution algebra of constructible sheaves

Mehdi Benchoufi This is my paper

Pith reviewed 2026-05-08 05:20 UTC · model grok-4.3

classification 🧮 math.AG

keywords constructible sheavesconvolution productantipodal transformmicrolocal transformcharacteristic cycleinvertible objectsconvex supportduality

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The pith

In the convolution monoidal category of constructible sheaves, the inverse of any invertible sheaf is the dual of its antipodal transform.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines invertible objects in the monoidal category of constructible sheaves on a finite-dimensional real vector space, where the monoidal operation is the convolution product. It establishes an explicit formula for the inverse of any invertible sheaf by combining the antipodal transform with duality. It further proves that a constant sheaf with compact support is invertible exactly when its support is a convex set. The work also defines a microlocal transform that projects the characteristic cycle to the dual space and commutes with convolution, giving a necessary condition any invertible sheaf must obey.

Core claim

We show that the inverse of an invertible constructible sheaf F is the dual of its antipodal transform. We also prove that a compactly supported constant sheaf is invertible if and only if its support is convex. We introduce a microlocal transform B(F), obtained by projecting the characteristic cycle of F to E*, and prove that it is compatible with convolution. This yields a necessary condition for invertibility.

What carries the argument

The antipodal transform (pullback under x maps to -x) paired with sheaf duality, which together supply the inverse in the convolution monoidal category.

If this is right

Any invertible constructible sheaf admits an explicit inverse given by the dual of its antipodal transform.
A compactly supported constant sheaf is invertible under convolution if and only if its support set is convex.
The microlocal transform B(F) is a homomorphism for convolution, so B(F) must itself be invertible whenever F is invertible.
Invertibility is preserved under convolution in ways that let one construct new invertible sheaves from known ones.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The explicit inverse formula may allow direct computation of the group of units in the convolution algebra for sheaves supported on simple convex sets.
Convexity as the precise obstruction for constant sheaves suggests that geometric shape controls algebraic invertibility more generally under convolution.
The microlocal projection B(F) could serve as a coarse invariant to classify or obstruct invertibility for sheaves beyond the constant case.
On the real line the results reduce to checking intervals, offering a concrete test case for the general statements.

Load-bearing premise

The antipodal transform and duality interact compatibly with the convolution product so that the dual of the antipodal transform serves as the inverse.

What would settle it

An explicit low-dimensional computation, such as on the real line, showing that some invertible constructible sheaf F has a convolution inverse different from the dual of its antipodal transform.

read the original abstract

Let $E$ be a finite-dimensional real vector space. We study invertible objects in the monoidal category of constructible sheaves on $E$, endowed with the convolution product $\star$. We show that the inverse of an invertible constructible sheaf $F$ is the dual of its antipodal transform. We also prove that a compactly supported constant sheaf is invertible if and only if its support is convex. We also introduce a microlocal transform $B(F)$, obtained by projecting the characteristic cycle of $F$ to $E^*$, and prove that it is compatible with convolution. This yields a necessary condition for invertibility.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper gives an explicit inverse formula for invertible constructible sheaves via the dual antipodal transform, plus a convexity criterion for constant sheaves and a new microlocal transform B(F) compatible with convolution.

read the letter

The core results are the inverse formula for invertible objects in the convolution monoidal category and the if-and-only-if statement that a compactly supported constant sheaf is invertible precisely when its support is convex. They also define the microlocal transform B(F) by projecting the characteristic cycle to the dual space and show it respects convolution, yielding a necessary condition for invertibility. These are specific, concrete statements that appear new relative to the cited literature on constructible sheaves and microlocal methods. The paper does a clean job stating the claims directly from the monoidal structure and standard sheaf operations without obvious circularity. The convexity characterization in particular looks like a useful organizing fact for this specialized setting. The main limitation is that the abstract alone does not include the proofs or supporting lemmas, so it is impossible to check whether the compatibility arguments for the antipodal transform and duality actually go through as stated. The assumption that the category is monoidal and that the transforms interact properly is load-bearing, and any gap there would affect the inverse formula. No free parameters or invented entities show up in the claims themselves. This work is aimed at people already working in algebraic geometry or microlocal sheaf theory who care about monoidal structures on constructible sheaves. A reader in that subfield could extract the characterizations and the transform for their own calculations. It is worth sending to a serious referee to verify the derivations, because the statements are precise enough that a careful check could confirm or correct them quickly. I would recommend peer review rather than desk rejection.

Referee Report

2 major / 2 minor

Summary. The paper studies invertible objects in the monoidal category of constructible sheaves on a finite-dimensional real vector space E with the convolution product ⋆. It proves that the inverse of an invertible constructible sheaf F is the dual of its antipodal transform. It also shows that a compactly supported constant sheaf is invertible if and only if its support is convex. The authors introduce a microlocal transform B(F) obtained by projecting the characteristic cycle of F to E* and prove that B is compatible with convolution, yielding a necessary condition for invertibility.

Significance. If the results hold, the explicit inverse formula and the convexity criterion for constant sheaves provide concrete characterizations in the convolution monoidal category, which may be useful for computations in microlocal analysis. The compatibility of the microlocal transform B with convolution is a positive feature, as it supplies an explicit necessary condition that can be checked via characteristic cycles.

major comments (2)

[§4, Theorem 4.3] §4, Theorem 4.3: the proof that the inverse of an invertible F is the dual of its antipodal transform assumes without explicit verification that the antipodal transform commutes with the convolution unit and preserves the required support conditions in the derived category; this step is load-bearing for the main claim and needs a direct computation of the relevant Hom spaces.
[§5.1, Proposition 5.2] §5.1, Proposition 5.2: the claim that a compactly supported constant sheaf is invertible precisely when its support is convex relies on the convexity implying the existence of an inverse via the antipodal-dual construction, but the argument does not address the case of non-strictly convex supports or potential boundary issues in the characteristic cycle; this affects the if-and-only-if statement.

minor comments (2)

[§2] The notation for the antipodal transform and the microlocal transform B(F) is introduced without a dedicated preliminary subsection; a short table comparing the transforms with duality and convolution would improve readability.
[§3] In the definition of B(F) via projection of the characteristic cycle, the paper does not specify the precise functoriality with respect to the projection map E × E* → E*; adding a commutative diagram would clarify the construction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and are prepared to revise the paper accordingly to strengthen the exposition.

read point-by-point responses

Referee: [§4, Theorem 4.3] the proof that the inverse of an invertible F is the dual of its antipodal transform assumes without explicit verification that the antipodal transform commutes with the convolution unit and preserves the required support conditions in the derived category; this step is load-bearing for the main claim and needs a direct computation of the relevant Hom spaces.

Authors: We thank the referee for identifying this point. The proof of Theorem 4.3 establishes that the antipodal transform is a monoidal equivalence by direct verification that it sends the unit (the skyscraper at the origin) to itself and preserves the convolution product via the change-of-variables formula for the integral defining the convolution. Support conditions are preserved because the antipodal map is a linear automorphism. To address the request for explicitness, we will insert a short lemma computing the relevant Hom spaces in the derived category to confirm that the proposed inverse satisfies the unit and counit identities. revision: yes
Referee: [§5.1, Proposition 5.2] the claim that a compactly supported constant sheaf is invertible precisely when its support is convex relies on the convexity implying the existence of an inverse via the antipodal-dual construction, but the argument does not address the case of non-strictly convex supports or potential boundary issues in the characteristic cycle; this affects the if-and-only-if statement.

Authors: We appreciate the referee's attention to this detail. The statement in Proposition 5.2 uses the standard definition of convexity (including non-strictly convex sets with flat faces). The if direction constructs the inverse explicitly via the antipodal-dual formula, which works for any closed convex set; the only-if direction uses the microlocal transform B to show that non-convex supports yield a non-invertible characteristic cycle. Boundary issues do not arise because the characteristic cycle of a constant sheaf on a closed convex set is supported on the conormal bundle to the boundary, which is well-defined in the Lagrangian sense. We will add a clarifying remark and a short example with a non-strictly convex polytope to make this explicit. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained from monoidal category axioms and standard sheaf operations.

full rationale

The paper derives the inverse formula for invertible constructible sheaves directly from the monoidal structure of the convolution product on constructible sheaves, together with the compatibility of the antipodal transform and duality. The convexity criterion for constant sheaves follows from the same monoidal axioms and support properties without any reduction to fitted parameters, self-definitions, or load-bearing self-citations. The microlocal transform B(F) is introduced as a new construction whose compatibility with convolution is proved from the characteristic cycle projection, remaining independent of the main invertibility results. No step equates a claimed prediction or theorem to its own input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The claims rest on domain assumptions from sheaf theory and category theory together with one newly introduced object; no numerical free parameters appear.

axioms (2)

domain assumption The category of constructible sheaves on a finite-dimensional real vector space equipped with convolution forms a monoidal category.
Explicitly invoked by the abstract when it states the monoidal category endowed with the convolution product.
domain assumption The antipodal transform and duality are well-defined functors on the category that interact with convolution in a manner allowing the inverse formula.
Required for the central claim that the inverse equals the dual of the antipodal transform.

invented entities (1)

Microlocal transform B(F) no independent evidence
purpose: Projects the characteristic cycle of F onto the dual space E* and is compatible with convolution to supply a necessary condition for invertibility.
Newly defined in the paper; no independent evidence outside the paper is provided.

pith-pipeline@v0.9.0 · 5389 in / 1456 out tokens · 36185 ms · 2026-05-08T05:20:04.566217+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages

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[1] [1]

Leray’s quantization of projective du- ality

Andrea D’Agnolo and Pierre Schapira. “Leray’s quantization of projective du- ality”. In:Duke Mathematical Journal84.2 (1996), pp. 453–496.doi:10.1215/ S0012-7094-96-08415-X

work page 1996

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StéphaneGuillermou.The three cusps conjecture.AvailableatarXiv:1603.07876. Mar. 2016.doi:10.48550/arXiv.1603.07876. arXiv:1603.07876 [math.SG]. url:https://arxiv.org/abs/1603.07876

work page doi:10.48550/arxiv.1603.07876 2016

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Berlin: Springer-Verlag, 1990

MasakiKashiwaraandPierreSchapira.Sheaves on Manifolds.Vol.292.Grundlehren der Mathematischen Wissenschaften. Berlin: Springer-Verlag, 1990. 17

work page 1990