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arxiv: 2409.14404 · v3 · submitted 2024-09-22 · 🧮 math.DG

Singularity formation in co-dimension one of the dHYM cotangent flow on blow up of mathbb{C}mathbb{P}³ at a point

Ramesh Mete This is my paper

Pith reviewed 2026-05-23 21:08 UTC · model grok-4.3

classification 🧮 math.DG
keywords dHYM cotangent flowsingularity formationblow-up of CP^3exceptional divisorCalabi ansatzsingular dHYM equationKähler geometryco-dimension one singularity
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The pith

The dHYM cotangent flow on the one-point blow-up of CP^3 develops a co-dimension one singularity along the exceptional divisor whose limit satisfies the singular dHYM equation.

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

The paper examines the dHYM cotangent flow on the blow-up of CP^3 at a point by restricting to the Calabi ansatz that preserves the natural symmetry. Under this reduction an explicit solution is constructed whose curvature blows up along the exceptional divisor while remaining controlled elsewhere. The resulting limit object satisfies the singular version of the dHYM equation previously defined for surfaces. This supplies a concrete three-dimensional instance supporting the conjecture that such singular solutions exist on manifolds with sufficient symmetry.

Core claim

An explicit example is constructed in which the dHYM cotangent flow on the one-point blow-up of CP^3 develops a singularity along the exceptional divisor. The flow is reduced via the Calabi ansatz, the singularity forms in co-dimension one, and the limit current satisfies the singular dHYM equation in the sense introduced in DMS24, thereby furnishing evidence for Conjecture 1.12 of that work on this symmetric three-dimensional manifold.

What carries the argument

The Calabi ansatz under the assumed symmetry on the blow-up of CP^3, which reduces the evolution to an ODE whose solution exhibits blow-up precisely along the exceptional divisor.

Load-bearing premise

The assumed symmetry together with the Calabi ansatz is sufficient to produce and capture the co-dimension one singularity along the exceptional divisor.

What would settle it

An explicit computation or numerical integration showing that the reduced ODE solution remains smooth across the exceptional divisor or that its limit fails to solve the singular dHYM equation.

read the original abstract

The existence and uniqueness of canonical singular solutions of the J-equation and the deformed Hermitian Yang Mills (dHYM) equation was proved in \cite{DMS24} on compact K\"{a}hler surfaces. In this paper, we study the singularity formation of the dHYM cotangent flow on the one-point blow up of $\mathbb{C}\mathbb{P}^3$ using Calabi ansatz. In particular, we provide an explicit example where the flow develops a singularity along the exceptional divisor. Moreover, the limit satisfies corresponding singular dHYM equation in the sense of \cite{DMS24} and provides some evidence for Conjecture $1.12$ in \cite{DMS24} on this three-dimensional manifold with symmetry.

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.

Referee Report

1 major / 0 minor

Summary. The paper studies singularity formation for the dHYM cotangent flow on the one-point blow-up of CP^3. Using the Calabi ansatz with symmetry, it claims an explicit example in which the flow develops a singularity along the exceptional divisor; the limit is asserted to satisfy the singular dHYM equation in the sense of DMS24 and to supply evidence for Conjecture 1.12 of DMS24 on this symmetric three-fold.

Significance. If the symmetry class is preserved and the ODE reduction is valid, the construction would supply a concrete, symmetry-reduced example of co-dimension-one singularity formation for the dHYM flow in dimension three, together with a limit that satisfies the singular equation of DMS24. This would constitute supporting evidence for the cited conjecture on a manifold with explicit symmetry.

major comments (1)
  1. [reduction to the ODE / main construction] The central construction reduces the flow to a scalar ODE via the Calabi ansatz on the blow-up, but the manuscript provides neither a derivation nor a citation establishing that the dHYM cotangent flow operator preserves the Calabi-symmetric class. Without this preservation result, the ODE does not necessarily capture the evolution of the full PDE, undermining the claim that the constructed limit satisfies the singular dHYM equation in the sense of DMS24.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on the manuscript. We respond to the major comment below.

read point-by-point responses

  1. Referee: [reduction to the ODE / main construction] The central construction reduces the flow to a scalar ODE via the Calabi ansatz on the blow-up, but the manuscript provides neither a derivation nor a citation establishing that the dHYM cotangent flow operator preserves the Calabi-symmetric class. Without this preservation result, the ODE does not necessarily capture the evolution of the full PDE, undermining the claim that the constructed limit satisfies the singular dHYM equation in the sense of DMS24.

    Authors: We agree that an explicit verification of the invariance of the Calabi-symmetric class under the dHYM cotangent flow is necessary to justify the reduction. In the revised manuscript we will add a dedicated subsection deriving this preservation: starting from the general expression of the dHYM cotangent flow operator, we substitute the Calabi ansatz (which depends only on the radial coordinate along the exceptional divisor and the base) and show by direct computation that all evolution equations remain within the ansatz class whenever the initial data does. This establishes that the ODE reduction is valid for symmetric initial metrics and therefore that the constructed limit satisfies the singular dHYM equation in the sense of DMS24. revision: yes

Circularity Check

0 steps flagged

No circularity: explicit ansatz-based example relies on external DMS24 definition and conjecture

full rationale

The paper constructs an explicit example of co-dimension one singularity formation for the dHYM cotangent flow on the one-point blow-up of CP^3 by imposing the Calabi ansatz and symmetry, then tracks the resulting reduced evolution to a singular limit along the exceptional divisor. This limit is asserted to satisfy the singular dHYM equation exactly as defined in the external reference DMS24 and to supply evidence for Conjecture 1.12 therein. No self-definitional steps, fitted parameters renamed as predictions, load-bearing self-citations, or ansatz smuggling appear in the abstract or described derivation chain. The construction is an independent explicit solution within the assumed symmetry class rather than a tautological reduction to the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no free parameters, axioms, or invented entities can be identified from the given text.

pith-pipeline@v0.9.0 · 5664 in / 1022 out tokens · 31427 ms · 2026-05-23T21:08:35.425974+00:00 · methodology

discussion (0)

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Reference graph

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