Notes on the deformed Hermitian-Yang-Mills equations and the large scaling limits of stability conditions

Yu-Wei Fan

arxiv: 2604.22246 · v1 · submitted 2026-04-24 · 🧮 math.AG · math.DG

Notes on the deformed Hermitian-Yang-Mills equations and the large scaling limits of stability conditions

Yu-Wei Fan This is my paper

Pith reviewed 2026-05-08 10:14 UTC · model grok-4.3

classification 🧮 math.AG math.DG

keywords deformed Hermitian-Yang-Mills equationslarge scaling limitsstability conditionsline bundlescomplex projective surfacesKähler formsArcara-Miles conjecture

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The pith

Assuming the Arcara-Miles conjecture, a line bundle on any smooth complex projective surface admits a deformed Hermitian-Yang-Mills metric exactly when it is stable in the large scaling limit for a generic Kähler form.

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

The paper proves an equivalence between two conditions on line bundles over smooth complex projective surfaces. Assuming the Arcara-Miles conjecture, existence of a deformed Hermitian-Yang-Mills metric is equivalent to stability in the large scaling limit with respect to a generic Kähler form. This extends a prior result that was known only when the surface is toric. A reader would care because the equivalence supplies an analytic criterion for an algebraic stability notion that appears in limiting regimes of stability conditions on derived categories.

Core claim

Assuming a conjecture of Arcara and Miles, a line bundle on a smooth complex projective surface admits a deformed Hermitian-Yang-Mills metric if and only if it is stable in the large scaling limit with respect to a generic Kähler form. The note observes that the same statement already proved by Stoppa for toric surfaces therefore holds for arbitrary smooth projective surfaces once the conjecture is granted.

What carries the argument

The Arcara-Miles conjecture, which supplies the missing bridge that lets the toric-surface equivalence extend to the general case by relating the deformed Hermitian-Yang-Mills equation directly to the large-scaling stability condition.

If this is right

The analytic condition of solving the deformed Hermitian-Yang-Mills equation becomes a practical test for the algebraic stability condition in the large scaling limit.
Stability of line bundles in the large scaling limit of a generic Kähler form can be decided by existence of a metric satisfying the deformed equation.
The equivalence removes the toric restriction that limited earlier work and applies uniformly to every smooth complex projective surface.

Where Pith is reading between the lines

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

If the conjecture holds, one could attempt to replace purely algebraic checks of large-scaling stability with numerical or analytic approximation schemes based on the deformed equation.
The result suggests that similar equivalences might be examined for higher-rank bundles or for stability conditions on threefolds once an appropriate analogue of the Arcara-Miles conjecture is available.
The large scaling limit may acquire a direct geometric meaning through the existence of these metrics, potentially linking it to other limiting regimes such as adiabatic limits in special Lagrangian geometry.

Load-bearing premise

The Arcara-Miles conjecture is true for the surfaces in question.

What would settle it

A single line bundle on a non-toric smooth projective surface that possesses a deformed Hermitian-Yang-Mills metric yet fails large-scaling stability (or vice versa) would disprove the claimed equivalence.

read the original abstract

In this short note, we show that, assuming a conjecture of Arcara and Miles, a line bundle on a smooth complex projective surface admits a deformed Hermitian-Yang-Mills metric if and only if it is stable in the ``large scaling limit" with respect to a generic K\"ahler form. The same statement for toric surfaces was recently proved by Stoppa. The purpose of this note is to remark that this equivalence holds for arbitrary smooth projective surfaces.

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

Short note observes that a conditional equivalence between dHYM metrics and large-limit stability extends from toric surfaces to general ones once the Arcara-Miles conjecture is granted, with no independent proof or new evidence supplied.

read the letter

This note's main point is that Stoppa's result for toric surfaces carries over to arbitrary smooth projective surfaces if you assume the Arcara-Miles conjecture. The author states the equivalence directly: a line bundle admits a deformed Hermitian-Yang-Mills metric precisely when it is stable in the large scaling limit for a generic Kähler form, under that assumption. The text stays explicit about the scope and the external hypothesis, which is the right way to handle it. The citation to Stoppa is accurate and the reduction does not introduce circularity or hidden assumptions inside the paper itself. If the conjecture is true, the generalization appears to hold without toric-specific tools, which is a clean observation. The math checks out on its own terms as far as the abstract and description go, with no internal contradictions or invented steps. The soft spot is that the entire claim rests on an unproven external conjecture, and the note supplies no new partial results, examples, or checks to support it. The contribution is therefore limited to pointing out that the toric case does not rely on toric geometry in an essential way. This is the kind of remark that specialists tracking stability conditions on surfaces might want to see, but it does not stand alone or advance the conjecture itself. For readers already working in this corner of algebraic geometry it could be worth a quick look to avoid duplicating the extension. I would not cite it in my own work until the conjecture is resolved, and I would not send it for full peer review; a short note slot might be appropriate if the journal wants it, but it does not need serious referee time.

Referee Report

0 major / 2 minor

Summary. The manuscript is a short note claiming that, assuming a conjecture of Arcara and Miles, a line bundle on a smooth complex projective surface admits a deformed Hermitian-Yang-Mills metric if and only if it is stable in the large scaling limit with respect to a generic Kähler form. This extends Stoppa's earlier result, which was restricted to toric surfaces.

Significance. If the Arcara-Miles conjecture holds, the note supplies a direct generalization of the dHYM existence/stability equivalence to all smooth projective surfaces. This removes the toric restriction while preserving the conditional character of the statement and citing the relevant prior work on toric surfaces. The contribution is modest but cleanly stated, and the explicit dependence on the external conjecture is a strength rather than a hidden weakness.

minor comments (2)

The abstract and introduction could briefly recall the precise statement of the Arcara-Miles conjecture (or give a reference to its exact formulation) so that readers need not consult the cited paper to understand the scope of the assumption.
Consider adding a short remark on whether the genericity condition on the Kähler form is inherited directly from Stoppa's toric argument or requires any additional verification when the conjecture is invoked.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. The report correctly captures the scope of the note as a conditional extension of Stoppa's toric result to arbitrary smooth projective surfaces under the Arcara-Miles conjecture.

Circularity Check

0 steps flagged

No significant circularity; central claim conditional on external conjecture

full rationale

The manuscript is a short note whose equivalence result is explicitly conditional on the Arcara-Miles conjecture (an external assumption) and extends Stoppa's prior independent result for toric surfaces to general smooth projective surfaces. No load-bearing step reduces by definition, by fitted input, or by self-citation chain to the paper's own inputs. The derivation chain invokes the conjecture to replace toric-specific analysis without introducing internal circularity or renaming known results as new derivations.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the Arcara-Miles conjecture as an external assumption and on the prior definition of large scaling limit stability; no free parameters or new entities are introduced in the abstract.

axioms (1)

domain assumption Arcara-Miles conjecture
Explicitly assumed to obtain the if-and-only-if equivalence for general surfaces.

pith-pipeline@v0.9.0 · 5365 in / 1213 out tokens · 65826 ms · 2026-05-08T10:14:10.550772+00:00 · methodology

Notes on the deformed Hermitian-Yang-Mills equations and the large scaling limits of stability conditions

Core claim

What carries the argument

If this is right

Where Pith is reading between the lines

Load-bearing premise

What would settle it

discussion (0)