{"id":"dd55bdeb-0070-42da-8716-8111538d235e","arxiv_id":"2501.15251","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper proves a Bogomolov-Gieseker type inequality for local P^3 and uses it to construct families of geometric stability conditions and boundary points of the geometric chamber.","lead":"On the total space of the canonical line bundle over P^3 (a local Calabi-Yau threefold), the paper constructs new stability conditions on the derived category of sheaves supported on the zero section, and identifies some boundary points of the geometric stability space. The result extends a conjectural framework of Bayer-Macrì-Toda to a new threefold, marking progress toward understanding stability on all threefolds.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The Macrì reduction (Prop. 3.30) is the load-bearing bridge from the small-ω inequality to all of U, but its proof contains an unresolved 'Proporsition ??' and an unproved Jordan–Hölder factor step; Theorem 4.5 also omits Prop. 3.34, leaving β∈(0,1/2] uncovered.","rationale":"The reader's weakest assumption identifies exactly the load-bearing gap in the proof of Theorem 4.5. The paper contains real independent support: the explicit central-charge computations in Prop. 4.4 are concrete and plausible, the exceptional-collection heart strategy is sound, and the O boundary construction in Section 5 is checked directly. However, the passage from the verified small-ω inequality to the full family rests on Prop. 3.30, whose printed proof has a broken internal reference and an asserted Jordan–Hölder factor step. These are not cosmetic issues: unless the first-wall existence is supplied and the JH factor is shown to inherit the violation while driving the descent, the theorem only proves the small-ω region. The additional omission of Prop. 3.34 from the proof of Theorem 4.5 is a separate but concrete defect, since the stated reductions do not cover β∈(0,1/2]. The theorem-statement typo involving Coh^β_0 versus Coh^{β,α}_0 is closely related and should also be corrected, but it is less load-bearing than the reduction gap. Because the gaps appear repairable rather than fatal, the appropriate verdict remains CONDITIONAL: the paper should not be rejected outright, but the central claim cannot be accepted until Prop. 3.30 and the full reduction chain are written out and checked.","tokens_in":57781,"tokens_out":16149,"duration_ms":150045,"concrete_test":"Take a rational point with β∈(0,1/2] and small ω, for example (β,α)=(1/4,0.1), and trace the proof chain of Theorem 4.5: check whether Prop. 4.4 plus Props 3.30, 3.32, and 3.33 reach this point, or whether Prop. 3.34 is indispensable. Then complete Prop. 3.30 by replacing 'Proporsition ??' with the precise wall-crossing statement (Prop. 3.20 or 3.21) and proving the Jordan–Hölder factor claim for E1; if either step cannot be completed, the only established region is the small-ω case.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The full family in Theorem 4.5 is obtained by combining the small-ω Bogomolov–Gieseker inequality of Prop. 4.4 with the global reduction of Prop. 3.30, but Prop. 3.30 is not proved as written. In the descent argument (Section 3.5), the existence of the first wall where E0 becomes strictly tilt-semistable is justified by an unresolved reference 'Proporsition ??', and the assertion that a Jordan–Hölder factor E1 of E0[l0] can be chosen with v^{β1}_3(E1) > ((2α1−β1^2)/6)v^{β1}_1(E1) is stated without proof. The violation can be recovered by additivity and the strict decrease of v^{β_{n+1}}_1(E_{n+1})^2 follows from positivity of v^β_1 on finite-slope objects in Coh^β_0, but these lemmas are not stated and the wall-crossing step is essential: without it the transition from a destabilizing chamber to a JH factor is not established. Separately, the proof of Theorem 4.5 cites only Props 3.30, 3.32, and 3.33, omitting Prop. 3.34. Since Prop. 4.4 covers only rational β∈[−1/2,0], the dualization reduction of Prop. 3.34 is needed to cover β∈(0,1/2]; as printed the cited chain does not cover this range. Thus the central claim that the constructed objects form geometric stability conditions for all (β,α)∈U is not yet established.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies Bridgeland stability conditions on the bounded derived category D^b_0(X) of coherent sheaves supported on the zero section of X = Tot(ω_{P^3}). The central construction follows the Bayer–Macrì–Toda program: it defines a double-tilt heart Coh^{β,α}_0(X), a central charge Z^{β,α,a}, and reduces the required Bogomolov–Gieseker type inequality to a small-ω statement. The main theorem (Theorem 4.5) claims that for all (β,α) with α > β²/2 and a > (2α−β²)/6, Z^{β,α,a} is a stability function on the double-tilt heart, giving a full family of geometric stability conditions. The paper also constructs some boundary stability conditions using tilting at an exceptional object i_*O, identifies them as lying on the boundary of the geometric chamber, and applies spherical twists to obtain further conditions. The arguments are detailed, and the explicit sector computations in Proposition 4.4 appear correct. However, several load-bearing steps in the reduction chain are not fully proved as written, notably the descent argument in Proposition 3.30 and the covering of the full (β,α)-range in Theorem 4.5.","tokens_in":58095,"tokens_out":5844,"duration_ms":56340,"significance":"If the reduction chain is completed, the main theorem would provide the first full family of geometric stability conditions on D^b_0(Tot(ω_{P^3})), a natural local-P³ analogue of the Bayer–Macrì picture for the local P². The paper also gives a plausible route to boundary points of the geometric chamber and to autoequivalence-generated stability conditions. The proof is not circular: the central inequality is checked against an explicit candidate heart B, and no fitted parameters are used. The manuscript contains explicit computations and a serious attempt at all necessary reductions, which makes the remaining gaps concrete and likely fixable. As it stands, however, the printed proof does not establish the theorem for the full claimed range.","major_comments":[{"comment":"The reduction from the Bogomolov–Gieseker inequality for all (β,α) ∈ U to the small-ω region is load-bearing, but its proof is incomplete as printed. In the descent argument, the existence of the first wall where E₀ becomes strictly tilt-semistable is justified by an unresolved reference 'Proporsition ??', and the subsequent assertion that one can choose a Jordan–Hölder factor E₁ of E₀[l₀] with v^{β₁}_3(E₁) > ((2α₁−β₁²)/6)v^{β₁}_1(E₁) is stated without proof. The strict decrease of v^{β_{n+1}}_1(E_{n+1})² and the boundedness argument also rely on inequalities that are not stated as lemmas. Since Theorem 4.5 uses Proposition 3.30 to pass from the small-ω case to all of U, this gap affects the central claim.","section":"§3.5, Proposition 3.30"},{"comment":"The proof of Theorem 4.5 cites Propositions 3.30, 3.32, and 3.33, but it does not cite Proposition 3.34. Since Proposition 4.4 covers only rational β ∈ [−1/2, 0] with small ω, the dualization reduction of Proposition 3.34 is necessary to cover β ∈ (0, 1/2]. As printed, the cited chain does not establish the Bogomolov–Gieseker inequality for those β, so the theorem's claim for all (β,α) ∈ U is not obtained. The missing step should be added and the covering of the full β-range verified explicitly.","section":"§4, Theorem 4.5"},{"comment":"The theorem states that Z^{β,α,a} is a stability function on the bounded heart Coh^β_0(X), but the construction in Conjecture 3.27 and the surrounding discussion require the double-tilt heart Coh^{β,α}_0(X). As printed, the statement does not match the object that is actually being studied: Coh^β_0(X) is the first tilt heart, independent of α, and the sector computations in Proposition 4.4 are carried out for the second tilt heart. This is not merely a notation issue, because the claimed family of geometric stability conditions is defined on Coh^{β,α}_0(X); the theorem statement should be corrected.","section":"§4, Theorem 4.5 and §1.4, Theorem 1.6"},{"comment":"The proof of Theorem 4.5 establishes, at best, the Bogomolov–Gieseker inequality for the relevant semistable objects, but the theorem asserts that Z^{β,α,a} is a stability function for arbitrary real (β,α), a, not only for the rational small-ω points treated in Proposition 4.4. The passage from the rational small-ω inequality to all real (β,α) requires the Harder–Narasimhan property and the support property for the double-tilt heart; Remark 4.6 defers this to a deformation argument in [BMS16, Section 8] and says no details are included. Since this passage is part of the main theorem, it should be proved or explicitly cited in the proof of Theorem 4.5.","section":"§4, Theorem 4.5 and §3.4–3.6"}],"minor_comments":[{"comment":"The text contains the unresolved cross-reference 'Proporsition ??'; it should be replaced by a precise statement and proof of the asserted wall-crossing fact.","section":"§3.5, Proposition 3.30"},{"comment":"The proof of stability of T(−2)[1] invokes 'Proposition ??' for the wall through Π(T(−2)[1]); this reference must be supplied.","section":"§4, Proposition 4.3"},{"comment":"Remark 5.10 is incomplete: it ends with 'There is a similar consequence when β > µ_2(E) and', and the sentence is cut off.","section":"§5, after Remark 5.10"},{"comment":"The proof of Proposition A.13 refers to 'Lemma ??' when identifying the relevant subcategory; this reference should be made precise.","section":"§Appendix A, Proposition A.13"},{"comment":"There are numerous typographical issues, including 'Propostion', 'Asuume', 'boudary', and the garbled display in §1.2; a careful editing pass is needed.","section":"Throughout"},{"comment":"In the display of Proposition 5.13, expressions such as 'v^β_3(E)v^β_1(E)' should be written as quotients v^β_3(E)/v^β_1(E) to avoid confusion.","section":"§5.2, Proposition 5.13"}],"recommendation":"major_revision","confidential_remarks":"This is a promising manuscript with a credible overall strategy and explicit computations, but the printed proof of the main theorem is not yet complete: the small-ω reduction has an unresolved reference and an unproved Jordan–Hölder factor step, and the proof of Theorem 4.5 omits the dualization reduction needed for positive β. These are technical gaps rather than signs of a false result, so major revision is appropriate. I would not recommend acceptance until the reduction chain is completed and the theorem statement is corrected to use the intended double-tilt heart."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Punchline: the first family of geometric stability conditions on D^b_0(Tot(ω_{P^3})) is a real and natural result, and the core mechanism—the exceptional-collection heart plus the sector computation in Prop 4.4—is honest, explicit work. But as written, Theorem 4.5 is not fully proved. The reduction to small ω (Prop 3.30) contains an unresolved cross-reference ('Proporsition ??') and a Jordan–Hölder factor step that is asserted without proof. The stress-test note says the step is recoverable by additivity and positivity, and I think that is plausible, but it is not in the paper. Separately, the proof of Theorem 4.5 cites Props 3.30, 3.32, 3.33 and omits Prop 3.34, which is needed to go from β∈[-1/2,0] to β∈(0,1/2]. That is a small fix—the proposition is proved in Appendix A—but as printed the chain does not cover the claimed range.\n\nWhat is actually new: the existence of geometric stability conditions on local P3 has not appeared in the literature (BMT14 did P3, BM11 local P2). The construction of boundary points via tilting with i_*O and the spherical-twist statement in Section 5 go beyond a routine translation. The paper is also honest about what is conditional: Conjecture 5.4 is flagged as such, and the O-case is verified directly. The explicit calculations in Prop 4.4 are checkable and appear correct.\n\nThe soft spots are proportionate. The biggest one is the proof gap in Prop 3.30. If that reduction fails, the small-ω inequality only gives a chamber, not the full family. The gap is likely fillable, but it is load-bearing. The second is the missing citation of Prop 3.34. The third is draft-level hygiene: broken references and typos throughout.\n\nWho should read this: anyone working on stability conditions on local Calabi-Yau threefolds, or on BG-type inequalities. The paper deserves a serious referee: the result is significant, the method is standard enough to verify, and the gaps are repairable. I would send it to review and ask the referee to check Prop 3.30 and the β-range coverage carefully.","headline":"A significant new result—geometric stability conditions on local P3—but the written proof has two load-bearing gaps that need repair before the full family statement is established.","tokens_in":58696,"tokens_out":3140,"would_cite":true,"duration_ms":29128,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14F08","14J32","14J60"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that the local P3 carries a full family of geometric stability conditions, via a Bogomolov-Gieseker-type inequality.","keywords":["stability conditions","derived categories","canonical bundle","local P3","Bogomolov-Gieseker inequality","tilting","geometric stability","spherical twists"],"falsifier":"Look for a $\\nu_{\\beta,\\alpha}$-semistable object $E\\in\\operatorname{Coh}^{\\beta}_0(X)$ with tilt-slope $\\beta$ inside the small-$\\omega$ region ($\\beta\\in[-1/2,0]$, $\\alpha>\\beta^2/2$, $\\sqrt{2\\alpha-\\beta^2}<1/2$) whose Chern character satisfies $v_3^\\beta(E)>(2\\alpha-\\beta^2)v_1^\\beta(E)/6$; because the proof reduces the full inequality to exactly this region, one such object would disprove the central claim, and an exhaustive check of that numerical inequality for candidate objects would settle it.","tokens_in":57513,"feed_emoji":"📐","tokens_out":11735,"duration_ms":100858,"temperature":0.7,"pith_summary":"This paper aims to prove that the derived category of coherent sheaves supported on the zero section of the total space of the canonical line bundle over $\\mathbb{P}^3$ admits a large family of geometric stability conditions. The argument hinges on a Bogomolov-Gieseker-type inequality involving the third Chern character: for tilt-semistable objects with tilt-slope $\\beta$, one must have $v_3^\\beta(E) \\le (2\\alpha-\\beta^2)v_1^\\beta(E)/6$. Proving this inequality for the local $\\mathbb{P}^3$ is the core of the paper, and it is what turns a conjectural two-step tilting construction into actual stability conditions. If the construction works, skyscraper sheaves of all closed points remain stable in the same phase, which is exactly what a geometric stability condition requires.","feed_headline":"Geometric stability conditions found for canonical bundle of P3","feed_subtitle":"Proves the required Bogomolov-Gieseker inequality and reaches boundary stability conditions by tilting an exceptional object.","key_machinery":"The load-bearing object is the double-tilt heart $\\operatorname{Coh}^{\\beta,\\alpha}_0(X)$: first tilt the abelian category of sheaves supported on the zero section by slope $\\mu$, obtaining $\\operatorname{Coh}^{\\beta}_0(X)$, then tilt again by the tilt-slope $\\nu_{\\beta,\\alpha}$, obtaining the heart on which the three-parameter central charge has nonnegative imaginary part. The Bogomolov-Gieseker-type inequality supplies the real part: on objects where the imaginary part vanishes, it forces $\\operatorname{Re} Z^{\\beta,\\alpha,a} \\le 0$, which is the condition for a stability function. To prove that inequality, the paper uses a second bounded heart $\\mathcal{B}$ generated by $i_*\\mathcal{O}(1)$, $i_*\\mathcal{O}[1]$, $i_*T(-2)[2]$, $i_*\\mathcal{O}(-1)[3]$, together with a supporting lemma (Lemma 3.40) that converts half-plane containment of $Z(\\mathcal{B})$ and non-membership of $F[2]$ into the desired inequality for a simple object $F[1]$. A relative derived dual functor and a tensor-twist reduction are used to restrict the inequality to $\\beta\\in[-1/2,0]$.","core_discovery":"On its own terms, the paper claims that for $X = \\operatorname{Tot}(\\omega_{\\mathbb{P}^3})$ and for arbitrary real parameters $\\beta,\\alpha,a$ with $\\alpha > \\beta^2/2$ and $a > (2\\alpha-\\beta^2)/6$, the central charge $$$Z^{{\\beta,\\alpha,a}}$(E) = -v_3^\\$\\beta$(E) + a v_1^\\$\\beta$(E) + \\sqrt{-1}\\left(v_2^\\$\\beta$(E) - (\\$\\alpha$ - \\$beta^{2}$/2)v_0^\\$\\beta$(E)\\right)$$ defines a stability function on the double-tilt heart $\\operatorname{Coh}^{\\beta,\\alpha}_0(X)$, giving a family of geometric stability conditions on $D^b_0(X)$. The printed Theorem 4.5 names the single-tilt heart $\\operatorname{Coh}^{\\beta}_0(X)$; the surrounding construction and Conjecture 3.27 identify the intended heart as the double tilt. The proof reduces the required Bogomolov-Gieseker inequality to the small-$\\omega$ region $\\sqrt{2\\alpha-\\beta^2}<1/2$ with $\\beta\\in[-1/2,0]$, and verifies it there using a bounded heart generated by powers of $\\mathcal{O}(1)$, the tangent sheaf, and $\\mathcal{O}(-1)$ pushed forward from $\\mathbb{P}^3$. The same method then tilts at the exceptional object $i_*\\mathcal{O}$ to produce algebraic stability conditions lying on the boundary of the geometric stability space.","pith_inferences":["Beyond the paper, the same double-tilt construction should produce a wall-and-chamber decomposition of the geometric chamber for local $\\mathbb{P}^3$; the walls should be controlled by objects in the exceptional collection $(\\mathcal{O}(-1),T(-2),\\mathcal{O},\\mathcal{O}(1))$.","The boundary construction tilts only at $i_*\\mathcal{O}$; a natural extension is to tilt at the other exceptional bundles in the collection, which would identify additional boundary points whenever the analogue of Conjecture 5.4 holds.","The reduction to small $\\omega$ used here is the same strategy that proves stability for abelian threefolds; if it can be made to work under Assumption 3.8 for other vector bundles, the existence question for geometric stability on local Calabi-Yau threefolds would be settled in a wider class than local $\\mathbb{P}^3$.","A reader checking the proof should first test the unresolved descent step: if the Jordan-Hölder factor violating the inequality cannot always be chosen, the full family may still exist but would need a different proof even in the small-$\\omega$ case."],"forward_implications":["The region $\\{(\\beta,\\alpha,a): \\alpha>\\beta^2/2,\\ a>(2\\alpha-\\beta^2)/6\\}$ embeds into the stability space $\\operatorname{Stab}_H(D^b_0(X))$, so the stability space of the local $\\mathbb{P}^3$ has a nonempty open geometric chamber.","Every skyscraper sheaf $k(y)$ is stable of the same phase in this family, so the constructed stability conditions are geometric in the sense of the paper.","The boundary points $(Z^{\\beta,\\beta^2,a},\\operatorname{Coh}^{\\beta,i_*\\mathcal{O}}_0(X))$ are algebraic stability conditions and lie on $\\partial\\operatorname{Stab}^{\\mathrm{geo}}_H(D^b_0(X))$.","Spherical twists move this boundary: the same stability conditions lie in the intersection of the geometric stability space and its translate by the spherical twist at $i_*\\mathcal{O}$.","If the two assumptions in Remark 4.6(2) hold for other locally free sheaves on $\\mathbb{P}^3$, the same construction yields a continuous embedding for those total spaces as well."],"supporting_citations":[{"why":"Supplies the two-step tilting construction and the conjectural Bogomolov-Gieseker-type inequality that the paper proves for the local P3.","marker":"[BMT14]"},{"why":"Provides the method, used in Proposition 3.30, of reducing the Bogomolov-Gieseker inequality to small values of omega.","marker":"[Mac14]"},{"why":"Constructs the bounded heart B from the exceptional collection (O(-1), T(-2), O, O(1)) that is used to verify the inequality.","marker":"[Bri05]"},{"why":"Contributes the wall-and-chamber and boundary techniques, including the lemma used to place the constructed stability conditions on the boundary of the geometric region.","marker":"[BM11]"},{"why":"Provides the quadratic-form formulation of the inequality and the deformation argument used to embed parameters into the stability space.","marker":"[BMS16]"},{"why":"Supplies the wall-and-chamber structure for tilt-stability that governs how semistability changes as the parameters move.","marker":"[FTV21]"},{"why":"Establishes that exceptional locally free sheaves on P3 are slope stable, which underlies the tilts used in the boundary construction.","marker":"[Zub90]"}],"fun_headline_variants":["Stability conditions on canonical P^3 bundle: geometric and algebraic","New geometric stability conditions for P^3 canonical bundle","P^3 canonical bundle gets geometric stability conditions","Tilting produces algebraic boundary stability conditions on P^3","Spherical twists yield new stability conditions on P^3 canonical bundle"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole family rests on the reduction claim that proving the Bogomolov-Gieseker inequality only for small $\\omega = \\sqrt{2\\alpha-\\beta^2}$ is enough to prove it for every allowed $(\\beta,\\alpha)$; the descent through walls in that reduction chooses a Jordan-Hölder factor that still violates the inequality, and that choice is cited to an unresolved reference rather than demonstrated.","fun_headline_variants_meta":{"raw":{"variants":["Stability conditions on canonical P^3 bundle: geometric and algebraic","New geometric stability conditions for P^3 canonical bundle","P^3 canonical bundle gets geometric stability conditions","Tilting produces algebraic boundary stability conditions on P^3","Spherical twists yield new stability conditions on P^3 canonical bundle"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000824,"raw_usage":{"total_tokens":3581,"prompt_tokens":900,"completion_tokens":2681,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":516,"completion_tokens_details":{"reasoning_tokens":2598}},"tokens_in":516,"tokens_out":2681,"duration_ms":15888,"temperature":1.0,"reasoning_tokens":2598,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T14:28:48.615206+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Look for a $\\nu_{\\beta,\\alpha}$-semistable object $E\\in\\operatorname{Coh}^{\\beta}_0(X)$ with tilt-slope $\\beta$ inside the small-$\\omega$ region ($\\beta\\in[-1/2,0]$, $\\alpha>\\beta^2/2$, $\\sqrt{2\\alpha-\\beta^2}<1/2$) whose Chern character satisfies $v_3^\\beta(E)>(2\\alpha-\\beta^2)v_1^\\beta(E)/6$; because the proof reduces the full inequality to exactly this region, one such object would disprove the central claim, and an exhaustive check of that numerical inequality for candidate objects would settle it.","supporting_citations":[],"review_version":1}