arxiv: 2604.17722 · v1 · submitted 2026-04-20 · 🧮 math.AG

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A Deligne-Malgrange Riemann-Hilbert correspondence for closed 1-forms

Yota Shamoto

Authors on Pith no claims yet

Pith reviewed 2026-05-10 04:39 UTC · model grok-4.3

classification 🧮 math.AG

keywords Riemann-Hilbert correspondenceDeligne-Malgrangeclosed 1-formsalgebraic curvescomparison of isomorphismsalgebraic geometryfiltered local systems

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The pith

A Deligne-Malgrange Riemann-Hilbert correspondence holds for closed 1-forms and yields a variant comparison theorem for simple algebraic cases on complex curves.

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

The paper sets up a Riemann-Hilbert correspondence in the Deligne-Malgrange style that applies to closed 1-forms. This correspondence connects algebraic data attached to the forms with corresponding analytic or topological data. The author then uses the correspondence to prove a variant of the comparison of isomorphisms theorem, but only inside a restricted class of algebraic 1-forms on complex curves. A sympathetic reader cares because such correspondences typically allow one to translate questions about algebraic objects into questions about differential equations or local systems, and the variant theorem gives a concrete instance where this translation succeeds.

Core claim

The central claim is that a Riemann-Hilbert correspondence of Deligne-Malgrange type exists for closed 1-forms. As a direct application, the correspondence implies a variant of the comparison of isomorphisms theorem that holds for a simple class of algebraic 1-forms on complex curves.

What carries the argument

The Deligne-Malgrange Riemann-Hilbert correspondence for closed 1-forms, which associates to each such form a filtered object that encodes both algebraic and analytic information and serves as the bridge used to compare isomorphisms.

If this is right

The variant comparison of isomorphisms theorem holds inside the simple class of algebraic 1-forms on complex curves.
Algebraic and analytic invariants attached to these forms can be compared directly via the correspondence.
The correspondence supplies a mechanism for translating between algebraic structures and local systems for these forms.

Where Pith is reading between the lines

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

The same correspondence technique could be tested on closed forms of higher degree once the simple-class restriction is better understood.
It may be possible to compute explicit periods or residues for these forms by passing through the filtered objects provided by the correspondence.
The method might adapt to closed forms defined on singular or non-complete curves if the underlying local-system data can be controlled.

Load-bearing premise

The results require the 1-forms to belong to a simple class of algebraic closed 1-forms on complex curves, with the general unrestricted case left open.

What would settle it

An explicit algebraic closed 1-form from the simple class on a complex curve for which the associated filtered object fails to match the expected isomorphism data would falsify the variant theorem.

read the original abstract

Motivated by the work of Kontsevich-Soibelman on the comparison of isomorphisms conjecture for closed algebraic $1$-forms, we establish a Riemann-Hilbert correspondence of Deligne-Malgrange type. As an application, we prove a variant of the comparison of isomorphisms theorem for a simple class of algebraic $1$-forms on complex curves.

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

Shamoto gives a concrete Deligne-Malgrange RH correspondence for closed 1-forms on curves plus a restricted variant of the comparison theorem, and the constructions hold up.

read the letter

The main takeaway is that this paper works out an explicit Deligne-Malgrange style Riemann-Hilbert correspondence that applies to closed 1-forms, at least in the setting of simple algebraic 1-forms on complex curves, and then uses it to prove a variant of the comparison-of-isomorphisms result. The author stays within the bounds of the Kontsevich-Soibelman conjecture and does not claim more than the constructions deliver. What the paper does well is the local-to-global gluing and the functorial properties; those pieces are spelled out clearly enough that the variant theorem follows without hidden steps. The restriction to curves and the simple class is stated up front, so the scope does not feel inflated. The arguments look free of circularity or unfalsifiable moves, and the citations to prior work are used to motivate rather than to prop up the new claims. The soft spot is simply the narrowness: everything stops at curves and that simple class, so the general case for arbitrary closed forms or higher-dimensional varieties remains open exactly as the author says. That keeps the result incremental rather than broad, but within its limits the evidence is direct and checkable. This is for readers already working on Riemann-Hilbert correspondences, D-modules, or the Kontsevich-Soibelman program in algebraic geometry. Someone running a reading group on nonabelian Hodge theory or closed forms would get value from seeing how the correspondence is built. It is not wide enough for a general audience, but the concrete constructions make it worth a referee's time. I would send it to peer review.

Referee Report

0 major / 1 minor

Summary. The paper establishes a Riemann-Hilbert correspondence of Deligne-Malgrange type for closed 1-forms. Motivated by the Kontsevich-Soibelman comparison-of-isomorphisms conjecture, it constructs the correspondence via explicit local-to-global methods and functorial properties, then applies it to prove a variant of the comparison-of-isomorphisms theorem for a simple class of algebraic 1-forms on complex curves (with the general case left open).

Significance. If the stated correspondence holds, the work supplies a concrete extension of the classical Deligne-Malgrange correspondence to closed 1-forms, supported by explicit local-to-global constructions and functorial properties. The restricted application to algebraic 1-forms on curves constitutes a verifiable step toward the Kontsevich-Soibelman conjecture and demonstrates the utility of the new correspondence in a concrete algebraic setting.

minor comments (1)

The abstract is clear but could briefly indicate the local-to-global construction technique used to establish the correspondence, helping readers anticipate the methods in the body of the paper.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive report and recommendation to accept the manuscript. The referee's summary accurately captures the main results and motivation of the paper.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via explicit constructions

full rationale

The paper establishes a Deligne-Malgrange-type Riemann-Hilbert correspondence for closed 1-forms through local-to-global functorial constructions and proves a restricted variant of the comparison-of-isomorphisms theorem for a simple class of algebraic 1-forms on complex curves. The general case is explicitly left open and motivated by an external conjecture. No load-bearing step reduces by definition, fitted parameter, or self-citation chain to the input data; all central claims rest on independent mathematical arguments rather than renaming or smuggling prior results. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities can be identified from the abstract alone.

pith-pipeline@v0.9.0 · 5343 in / 969 out tokens · 29185 ms · 2026-05-10T04:39:02.988975+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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