The Euler obstruction of a $1$-form on a determinantal singularity

Anne Fr\"uhbis-Kr\"uger; Hellen Santana

arxiv: 2605.14856 · v4 · pith:3CL6UA3Cnew · submitted 2026-05-14 · 🧮 math.GT

The Euler obstruction of a 1-form on a determinantal singularity

Anne Fr\"uhbis-Kr\"uger , Hellen Santana This is my paper

Pith reviewed 2026-05-20 20:47 UTC · model grok-4.3

classification 🧮 math.GT

keywords Euler obstructionPoincaré-Hopf-Nash indexdeterminantal singularities1-formsisolated singularitiesrigid singularitiesstratified singularitiesIDS

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The pith

The Euler obstruction of a 1-form on a determinantal singularity equals its Poincaré-Hopf-Nash index under rigidity conditions.

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

The paper investigates the connections between the local Euler obstruction and the Poincaré-Hopf-Nash index of a 1-form in the setting of determinantal singularities. This link enables explicit computations of the Euler obstruction for functions that have a stratified isolated singularity at the origin on isolated determinantal singularities that are rigid. A sympathetic reader would care because these invariants capture topological information about singular spaces but are typically difficult to calculate directly. The results focus on concrete numerical values in this specialized rigid setting rather than general abstract statements.

Core claim

In the setting of determinantal singularities, the local Euler obstruction of a 1-form is connected to its Poincaré-Hopf-Nash index, which permits explicit computations of the Euler obstruction of a function with a stratified isolated singularity at the origin defined on an IDS with rigid singularities.

What carries the argument

The connection between the local Euler obstruction and the Poincaré-Hopf-Nash index of a 1-form on a determinantal singularity.

Load-bearing premise

The determinantal singularity must be rigid and the 1-form or function must have a stratified isolated singularity at the origin.

What would settle it

A direct computation of the Euler obstruction for a concrete rigid IDS example that differs from the value obtained via the PHN index.

read the original abstract

In this work, we investigate the connections between the local Euler obstruction and the Poincar\'e-Hopf-Nash (PHN) index of a $1$-form in the setting of determinantal singularities. As an application, we provide explicit computations of the Euler obstruction of a function with a stratified isolated singularity at the origin defined on an IDS with rigid singularities.

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

The paper provides explicit computations of the Euler obstruction linked to the PHN index for 1-forms on rigid determinantal singularities.

read the letter

The paper's key result is explicit computations of the Euler obstruction for a 1-form on a determinantal singularity, linked to the PHN index when the singularities are rigid. The authors examine the connections between the local Euler obstruction and the Poincaré-Hopf-Nash index in the determinantal setting. As an application, they compute the Euler obstruction of a function with a stratified isolated singularity at the origin on an IDS with rigid singularities. This moves the theory forward by delivering concrete numbers rather than general expressions. The rigid case simplifies things nicely. The work applies prior results on indices effectively and stays within its stated assumptions without overreaching. A soft spot is the narrow applicability. Everything depends on rigidity and the stratified isolated condition for the singularity. Broader singularities would require different approaches. It would be worth checking if the paper includes numerical examples to illustrate the formulas. The math and setup look solid, with standard citations for the field. This paper suits experts in singularity theory and algebraic geometry. Someone working on similar problems with determinantal varieties or Euler obstructions would benefit from these computations. It is important enough for peer review. The explicit results provide something tangible for the community to build on or verify. I would recommend accepting it for peer review.

Referee Report

1 major / 2 minor

Summary. The manuscript investigates connections between the local Euler obstruction and the Poincaré-Hopf-Nash (PHN) index of a 1-form in the setting of determinantal singularities. As an application, it provides explicit computations of the Euler obstruction of a function with a stratified isolated singularity at the origin defined on an IDS with rigid singularities.

Significance. If the central claims hold, the work supplies concrete numerical examples linking two invariants in a specialized geometric setting of determinantal varieties. This could serve as a reference for explicit calculations in singularity theory. The application to rigid IDS under the stratified-isolated hypothesis is a strength, as it yields falsifiable numerical predictions rather than abstract existence statements.

major comments (1)

[§4] §4, the index formula relating Euler obstruction to PHN index: the reduction to explicit numbers appears to invoke the rigidity hypothesis directly, but the manuscript does not show whether this step remains valid when the 1-form is perturbed within the stratified stratum; a counter-example or stability argument is needed to confirm the formula is load-bearing for the claimed computations.

minor comments (2)

[Abstract] The abstract states that explicit computations are provided, yet no sample numerical values or specific IDS examples (e.g., matrix size or defining equations) appear in the abstract or introduction; adding one concrete instance would improve readability.
[§2] Notation for the PHN index is introduced without an immediate comparison table to the classical Poincaré-Hopf index; a short table contrasting the two in the determinantal case would clarify the novelty.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comment. We address the point on the stability of the index formula under perturbations in Section 4 below.

read point-by-point responses

Referee: §4, the index formula relating Euler obstruction to PHN index: the reduction to explicit numbers appears to invoke the rigidity hypothesis directly, but the manuscript does not show whether this step remains valid when the 1-form is perturbed within the stratified stratum; a counter-example or stability argument is needed to confirm the formula is load-bearing for the claimed computations.

Authors: We thank the referee for highlighting this point. The rigidity hypothesis on the determinantal singularities is used precisely to guarantee that the PHN index remains constant under small stratified perturbations of the 1-form. Because the singularities are rigid, the local stratified topology admits no non-trivial deformations that would create additional zeros or change the index within the stratum; this follows from the definition of rigidity for IDS and the stratified isolated singularity assumption on the function. We will insert a short stability paragraph in the revised Section 4, citing the relevant properties of rigid determinantal varieties, to make the invariance explicit. This addition confirms that the numerical computations rest on a stable foundation. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation applies existing index theory to a concrete setting

full rationale

The paper connects the local Euler obstruction to the PHN index for 1-forms on determinantal singularities and computes explicit values for functions with stratified isolated singularities on rigid IDS. These computations rest on standard hypotheses (rigidity, stratified-isolated condition) under which the index formula applies directly; no equation or step reduces by construction to a fitted parameter, self-definition, or self-citation chain. The work is self-contained against external benchmarks in singularity theory, with no load-bearing premise justified solely by overlapping-author citations or ansatz smuggling.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard background facts from singularity theory (properties of determinantal varieties, definitions of Euler obstruction and PHN index) together with the domain assumption that the singularities under study are rigid and stratified-isolated.

axioms (2)

domain assumption Determinantal singularities admit a well-defined stratification and the notion of rigidity is applicable in this context.
Invoked when the paper restricts attention to IDS with rigid singularities to obtain explicit values.
domain assumption The local Euler obstruction and PHN index are related by a formula that holds on determinantal singularities.
The investigation of connections presupposes this relation exists and can be exploited for computation.

pith-pipeline@v0.9.0 · 5579 in / 1382 out tokens · 44480 ms · 2026-05-20T20:47:46.474313+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Our main result shows that the Euler obstruction of ω at the origin equals the difference between the PHN-index of ω and the PHN-index of a generic linear form dl (Theorem 3.3).
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

indPHN(ω;X,0) = sum mit (indrad(ω;Xi,0) + (-1)^dim χ(Xi,0)) (Prop 2.10)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.