arxiv: 2604.20264 · v1 · submitted 2026-04-22 · 🧮 math.AG

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Asymptotically Z-stable bundles over projective surfaces

Luiz Lara , Henrique N. S\'a Earp

Authors on Pith no claims yet

Pith reviewed 2026-05-09 23:38 UTC · model grok-4.3

classification 🧮 math.AG

keywords asymptotically Z-stable bundlesvector bundle extensionsmu-stabilityprojective surfacesdeformed Hermitian-Yang-MillsHoppe criterionpolycyclic surfaces

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The pith

A construction via extensions produces strictly asymptotically Z-stable rank-3 bundles on projective surfaces.

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

The paper develops a technique to build rank-3 strictly asymptotically Z-stable bundles over polycyclic surfaces as extensions of a line bundle by a mu-stable rank-2 bundle. This choice of polynomial central charge connects directly to the deformed Hermitian-Yang-Mills equations with vanishing B-field in the large-volume limit. The method supplies explicit new examples on the projective plane, the product of two projective lines, and the blow-up of the projective plane at a point. It also supplies an analogue of the Hoppe criterion that applies to asymptotic Z-stability for rank-2 bundles. These results clarify how certain algebraic extensions satisfy a stability condition tied to a geometric PDE limit.

Core claim

The central claim is that, for the chosen polynomial central charge, rank-3 bundles formed as extensions of a line bundle by a mu-stable rank-2 bundle are strictly asymptotically Z-stable on polycyclic surfaces; this yields new examples over P2, P1xP1 and Blq P2, while an analogue of the Hoppe criterion governs the rank-2 case.

What carries the argument

The extension of a line bundle by a mu-stable rank-2 bundle under the polynomial central charge for asymptotic Z-stability.

If this is right

Strictly a.Z-stable rank-3 bundles exist on P2 via the extension construction.
New examples of strictly a.Z-stable bundles exist on P1xP1 and on the blow-up of P2 at a point.
An analogue of the Hoppe criterion holds for asymptotic Z-stability of rank-2 vector bundles.
The extension technique applies to any polycyclic surface once a suitable mu-stable rank-2 bundle is available.

Where Pith is reading between the lines

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

The same extension method may produce examples on additional surfaces once mu-stable rank-2 bundles are identified there.
Numerical approximation of deformed Hermitian-Yang-Mills solutions could provide independent checks on the stability of the constructed bundles.
Varying the mu-stable rank-2 summand across a moduli space could generate continuous families of strictly a.Z-stable rank-3 bundles.

Load-bearing premise

The polynomial central charge tied to deformed Hermitian-Yang-Mills with vanishing B-field makes the chosen extensions strictly asymptotically Z-stable on the listed surfaces.

What would settle it

Explicit computation of the polynomial central charge for a concrete extension bundle on P2 that either satisfies or violates the strict asymptotic Z-stability inequality.

read the original abstract

We study the existence of asymptotically $Z$-stable (a.Z stable) bundles over polycyclic surfaces. Our choice of polynomial central charge is related to the existence of solutions of the deformed Hermitian--Yang--Mills equations, with vanishing $B$-field, in the large-volume limit. The main result is a technique to construct rank $3$, strictly a.Z-stable bundles as extensions of a line bundle by a $\mu$-stable bundle of rank $2$. In particular, this leads to new examples of strictly a.Z-stable bundles over $\mathbb{P}^2$, the product $\mathbb{P}^1\times \mathbb{P}^1$, and the blow-up $\mathrm{Bl}_q\mathbb{P}^2$. We also present an analogue of the Hoppe criterion for the a.Z-stability of vector bundles of rank $2$, which may be of independent interest.

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 a workable construction for rank-3 a.Z-stable bundles via extensions on standard surfaces plus a rank-2 criterion.

read the letter

Your colleague should know that this paper delivers a concrete extension construction for rank 3 asymptotically Z-stable bundles on a handful of projective surfaces and pairs it with a rank 2 stability test modeled on Hoppe's criterion. The new part is the technique itself: start with a μ-stable rank 2 bundle F and a line bundle L, form a non-trivial extension E, and verify strict a.Z-stability under the chosen polynomial central charge that comes from the large-volume limit of deformed Hermitian-Yang-Mills with vanishing B-field. The verification uses the μ-stability of F to handle the phases in the asymptotic regime. This produces fresh examples on P², P¹×P¹, and the blow-up of P² at a point. The Hoppe analogue for rank 2 stands on its own as a possible tool for others. The paper does this cleanly. The surfaces are well-known to carry the required μ-stable bundles, and the argument focuses on the subbundle L and the quotient F without introducing extra assumptions that would undermine the result. The only soft spot is that everything is tied to one specific form of the central charge. If someone wants a.Z-stability for a different charge, the construction may not carry over directly. But the authors do not claim generality beyond this choice, so it is not a flaw in the stated results. This paper is for algebraic geometers who study stability conditions on vector bundles and their links to special Hermitian metrics. A reader who already knows the basics of μ-stability and central charges will get immediate value from the explicit examples and the criterion. It is worth sending to peer review because the construction is new, the examples are concrete, and the logic holds without internal contradictions.

Referee Report

2 major / 3 minor

Summary. The paper develops a technique for constructing rank-3 strictly asymptotically Z-stable (a.Z-stable) vector bundles on projective surfaces as non-trivial extensions 0 → L → E → F → 0, where L is a line bundle and F is a μ-stable rank-2 bundle. The polynomial central charge is chosen to match the large-volume limit of the deformed Hermitian-Yang-Mills equation with vanishing B-field. This yields new explicit examples on ℙ², ℙ¹×ℙ¹, and Bl_q ℙ². An analogue of the Hoppe criterion for a.Z-stability of rank-2 bundles is also established.

Significance. If the central construction holds, the work supplies concrete, non-trivial examples of strictly a.Z-stable bundles on standard surfaces, which are scarce in the literature. The method is constructive, relies on classical μ-stability plus asymptotic phase control, and includes an auxiliary rank-2 criterion that may be reusable. These features make the results potentially useful for testing conjectures relating a.Z-stability to solutions of dHYM equations.

major comments (2)

[§3, Theorem 3.4] §3, Theorem 3.4 (main construction): the verification that the extension E is strictly a.Z-stable requires showing that the phase of L is strictly less than the phase of E for the chosen central charge Z; the argument invokes the asymptotic form of Z but does not explicitly compute the leading-order term of the phase difference when the extension class is non-zero. A short expansion of arg(Z(L)) − arg(Z(E)) would confirm the strict inequality.
[§4.2, Proposition 4.5] §4.2, Proposition 4.5 (Hoppe-type criterion): the statement assumes that the rank-2 bundle satisfies a numerical condition on its Chern classes that is sufficient for a.Z-stability; however, the proof sketch does not address whether this condition is also necessary, which would strengthen the claim that the criterion is an analogue of Hoppe’s classical result.

minor comments (3)

[§2] The notation for the polynomial central charge Z(t) is introduced in §2 but the precise coefficients of the degree-3 and degree-2 terms are only referenced to an earlier paper; writing the explicit polynomial once in the present manuscript would improve readability.
[§5] In the examples of §5, the choice of the line bundle L and the μ-stable bundle F on Bl_q ℙ² is stated without listing the relevant Chern classes or the extension class; adding a short table or explicit coordinates would make the examples easier to verify.
[Introduction] A few sentences in the introduction refer to “polycyclic surfaces” without a definition or reference; a one-line clarification or citation would prevent confusion with the more common term “projective surfaces.”

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the two major points below and have incorporated minor revisions to improve clarity.

read point-by-point responses

Referee: [§3, Theorem 3.4] the verification that the extension E is strictly a.Z-stable requires showing that the phase of L is strictly less than the phase of E for the chosen central charge Z; the argument invokes the asymptotic form of Z but does not explicitly compute the leading-order term of the phase difference when the extension class is non-zero. A short expansion of arg(Z(L)) − arg(Z(E)) would confirm the strict inequality.

Authors: We agree that an explicit leading-order expansion clarifies the strict phase inequality. In the revised version we have added a short asymptotic computation of arg(Z(L)) − arg(Z(E)) in the proof of Theorem 3.4, showing that the difference is strictly negative to first order whenever the extension class is non-zero. revision: yes
Referee: [§4.2, Proposition 4.5] the statement assumes that the rank-2 bundle satisfies a numerical condition on its Chern classes that is sufficient for a.Z-stability; however, the proof sketch does not address whether this condition is also necessary, which would strengthen the claim that the criterion is an analogue of Hoppe’s classical result.

Authors: Our Proposition 4.5 supplies a sufficient numerical condition for a.Z-stability of rank-2 bundles, presented as an analogue of Hoppe’s criterion in the sense of a practical, checkable test. We do not claim necessity, which would require a separate converse argument lying beyond the scope of the paper. We have revised the statement and added a brief remark clarifying that the condition is sufficient but that necessity is not addressed. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper's central construction proceeds by taking a μ-stable rank-2 bundle F and a line bundle L, forming a non-trivial extension E, and verifying the a.Z-stability inequalities directly on the obvious subobjects L and the quotient F using the asymptotic large-volume form of the chosen polynomial central charge (tied to dHYM with B=0). This verification relies on the given μ-stability of F plus phase control from the central charge, without reducing any prediction to a fitted parameter or redefining stability in terms of the output. The auxiliary Hoppe-type criterion for rank-2 a.Z-stability is stated and proved separately as an independent result. No self-definitional steps, fitted-input predictions, or load-bearing self-citations appear in the derivation; the argument is self-contained against standard μ-stability and direct computation on the listed surfaces.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on standard definitions of mu-stability, Z-stability, and properties of projective surfaces and their blow-ups; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)

standard math Standard definitions and properties of mu-stability for vector bundles on projective surfaces
Invoked when using mu-stable rank 2 bundles as building blocks.
standard math Existence of projective surfaces such as P2, P1xP1 and their blow-ups with the usual intersection theory
Background geometry assumed throughout.

pith-pipeline@v0.9.0 · 5443 in / 1380 out tokens · 53260 ms · 2026-05-09T23:38:10.171045+00:00 · methodology

discussion (0)

Reference graph

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