{"id":"58547ba5-8b73-4612-a92d-3ebc8867733c","arxiv_id":"2604.20264","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"A new extension-based construction produces strictly asymptotically Z-stable rank 3 bundles on polycyclic projective surfaces, plus a Hoppe-style criterion for rank 2 bundles.","lead":"The paper develops a construction technique for rank 3 asymptotically Z-stable vector bundles on projective surfaces by extending a line bundle with a mu-stable rank 2 bundle. This yields new examples on P2, P1xP1 and its blow-up, while linking to solutions of deformed Hermitian-Yang-Mills equations in the large-volume limit.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest assumption correctly isolates the dependence on the central-charge choice. Because the paper supplies an explicit construction and the required μ-stable bundles exist on the listed surfaces, the only remaining question is whether the chosen Z indeed forces strict a.Z-stability; the proposed check directly tests that step without assuming extra structure.","tokens_in":1664,"tokens_out":361,"duration_ms":24424,"concrete_test":"For the explicit P² example in §4, fix the chosen L and F, form E, and recompute the leading terms of the polynomial central charge Z(E) and Z(L) in the large-volume parameter; verify that the phase inequality Im(Z(L)/Z(E)) > 0 holds strictly and that no other rank-1 or rank-2 subsheaf violates it (using the explicit Chern classes given in the paper).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is a constructive technique producing rank-3 strictly a.Z-stable bundles via non-trivial extensions 0 → L → E → F → 0, with F μ-stable of rank 2, under a specific polynomial central charge tied to large-volume dHYM (B=0). The argument proceeds by verifying the stability inequality directly on the obvious subobjects (L and the quotient) and invoking the μ-stability of F plus the asymptotic form of Z to control phases. No internal contradiction appears in the setup; the surfaces (P², P¹×P¹, Bl_q P²) are standard and admit the required μ-stable rank-2 bundles. The Hoppe-type criterion for rank 2 is presented as an auxiliary result and does not affect the main construction.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","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.","tokens_in":1831,"tokens_out":669,"duration_ms":55810,"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":[{"comment":"§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.","section":"§3, Theorem 3.4"},{"comment":"§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.","section":"§4.2, Proposition 4.5"}],"minor_comments":[{"comment":"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.","section":"§2"},{"comment":"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.","section":"§5"},{"comment":"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.”","section":"Introduction"}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"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.","responses":[{"response":"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_made":"yes","referee_comment":"[§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."},{"response":"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_made":"yes","referee_comment":"[§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."}],"tokens_in":1389,"tokens_out":411,"duration_ms":17474,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","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.","headline":"This paper gives a workable construction for rank-3 a.Z-stable bundles via extensions on standard surfaces plus a rank-2 criterion.","tokens_in":2301,"tokens_out":416,"would_cite":false,"duration_ms":21799,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"A construction via extensions produces strictly asymptotically Z-stable rank-3 bundles on projective surfaces.","keywords":["asymptotically Z-stable bundles","vector bundle extensions","mu-stability","projective surfaces","deformed Hermitian-Yang-Mills","Hoppe criterion","polycyclic surfaces"],"falsifier":"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.","tokens_in":2575,"feed_emoji":"","tokens_out":673,"duration_ms":17658,"temperature":0.7,"pith_summary":"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.","feed_headline":"Extensions produce strictly a.Z-stable rank-3 bundles on surfaces","feed_subtitle":"The construction uses mu-stable rank-2 bundles plus line bundles and yields concrete examples on P2, P1xP1 and blow-ups.","key_machinery":"The extension of a line bundle by a mu-stable rank-2 bundle under the polynomial central charge for asymptotic Z-stability.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Extensions yield strictly a.Z-stable rank-3 bundles on projective surfaces","Strictly a.Z-stable rank-3 bundles from extensions on P2, P1xP1 and blow-ups","Hoppe criterion analogue for a.Z-stability of rank-2 bundles on surfaces","Rank-3 a.Z-stable bundles constructed via extensions over projective surfaces"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Extensions yield strictly a.Z-stable rank-3 bundles on projective surfaces","Strictly a.Z-stable rank-3 bundles from extensions on P2, P1xP1 and blow-ups","Hoppe criterion analogue for a.Z-stability of rank-2 bundles on surfaces","Rank-3 a.Z-stable bundles constructed via extensions over projective surfaces"]},"model":"grok-4.3","cost_usd":0.014223,"raw_usage":{"total_tokens":6017,"prompt_tokens":602,"num_sources_used":0,"completion_tokens":86,"cost_in_usd_ticks":142228000,"prompt_tokens_details":{"text_tokens":602,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":5329,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":602,"tokens_out":86,"duration_ms":26531,"temperature":1.0,"reasoning_tokens":5329,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-09T23:38:10.171045+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"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.","supporting_citations":[],"review_version":1}