{"id":"6c59ca6b-eb09-47fe-8d04-9d140b476de2","arxiv_id":"2506.21995","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A quotient of the stability manifold by an equivalence relation is a real manifold of half dimension, and the original manifold can be reconstructed from the quotient together with the relation ≲.","lead":"This paper introduces a way to halve the dimension of the space of Bridgeland stability conditions by identifying those that share the same imaginary central charge and lie in the same connected fiber. It proves the reduced space is a real manifold and that the original space can be rebuilt from it, then uses this to show how stability conditions on a variety would force stability conditions on all its smooth subvarieties.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the reconstruction argument survives scrutiny; the remaining delicate spot is the algebraic core of Proposition 4.5, but no concrete failure was found.","rationale":"The reader identified the most delicate part of the argument, namely the combination of Lemma 2.17, Proposition 2.16, and Proposition 4.5. I examined that core in detail and found the reasoning internally coherent. The quadratic-form lemma is computationally sound, the deformation argument respects the required negative-definiteness conditions, and the scaling step in Proposition 4.5 is justified by the valid shear action in Lemma A.4. The conditional geometric theorem is clearly labelled as depending on a conjecture that the authors themselves expect to fail in some threefolds, so this is a scope limitation rather than a flaw. Since I could not identify a concrete failure in the central structural claim, the appropriate verdict is unchanged, with the caveat that an independent computational check of Proposition 4.5 in the simplest nontrivial example would further de-risk the reconstruction theorem.","tokens_in":76352,"tokens_out":55982,"duration_ms":621149,"concrete_test":"As an independent verification, implement the curve case explicitly: for C of genus at least 1, take the known Stab(C)=sigma*fGL+(2,R) and compute Ta(sigma_tilde_t) and the relation lesssim from the explicit hearts; then check Corollary 4.8 by sampling 10^4 random (sigma_tilde,h) and testing whether h in Ta(sigma_tilde) is equivalent to the open two-sided lesssim condition in (4.12), with h perturbed in a small ball. If any sampled h lies in the lesssim-set but not in Ta (or vice versa), Proposition 4.5 fails and the reconstruction homeomorphism collapses.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I read the structural core in good faith. Corollary 4.8 rests on Proposition 4.5, whose proof invokes Lemma 2.17 and Lemma 4.6. I checked the quadratic-form construction in Lemma 2.17: the signature count, the negativity on the deformed kernels, and the inclusion neg(Qtilde) subset M_d union neg(Q) all follow from the displayed inequalities once epsilon is chosen small and Dtilde_1 is chosen large. The support-property claim inside Proposition 2.16 likewise uses the negative-definiteness on Ker(Z+tZ') and the contradiction with M_d, and the heart-constancy along the normalized deformation follows from Lemmas 2.9 and 2.10. Proposition 4.5's Main Case I.2 deforms along the pencil f+t(h0-sf) and reaches h0/s; the final scaling h0=s*(h0/s) is the valid d=0 case of Lemma A.4, so the algebra is sound. I tested the logic against the curve example Sb*(C): the direction h0=rk satisfies the one-sided lesssim condition but is also in Ta by the half-shift, and the two-sided condition defining TaSb correctly excludes would-be counterexamples. The geometric application (Theorem 1.3) is explicitly conditional on Conjecture 1.1/Stabd, and the paper states that the conjecture is expected to fail in some threefolds; this limits scope but is not an internal inconsistency. No load-bearing concern survived scrutiny.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces an equivalence relation on the manifold of Bridgeland stability conditions, identifies the quotient Sb(T) with a space of 'reduced stability conditions', and proves that Sb(T) carries a real (possibly non-Hausdorff) manifold structure of half the dimension of Stab(T). The main unconditional structural result is Corollary 4.8, which reconstructs Stab(T) from Sb(T) together with the relation ≲ via the subspace TaSb(T) of the tangent bundle. The paper then specializes to D^b(X) for smooth polarized varieties, gives explicit descriptions of reduced stability conditions for curves, surfaces, and threefolds, formulates Conjectures 1.1, 1.12, and 8.1, and proves that the Stabd and Sbd conjectures are equivalent. Theorem 1.3 derives geometric and numerical consequences, including the existence of stability conditions on all smooth subvarieties, conditional on Conjecture 1.1 or its weaker Stabd variant.","tokens_in":76669,"tokens_out":11747,"duration_ms":146032,"significance":"If the structural results are correct, the paper provides a genuinely new organizing principle for Bridgeland stability manifolds: the wall-and-chamber structure is preserved under reduction, and the original manifold is recovered from a real half-dimensional quotient together with a simple order-like relation. The proof of the reconstruction theorem is elaborate and rests on a substantial algebraic input (Lemma 2.17 and Proposition 4.5), and I checked the stress-test concern about Lemma 2.17 and Proposition 4.5 without finding a concrete failure. The geometric applications are explicitly conditional, and the paper is commendably honest about the expected failure of Conjecture 1.1 for some threefolds; this limits the unconditional scope but is not an internal inconsistency. The explicit examples, the treatment of the Bayer vanishing lemma, and the restriction theorem add significant value.","major_comments":[],"minor_comments":[{"comment":"The phrase 'an equivalent relation' appears repeatedly and should be 'an equivalence relation'; this occurs in Definition 2.11, in the paragraph after it, and in the introduction to Section 2.3.","section":"Definition 2.11 and Section 2.3"},{"comment":"The proof of Lemma 4.6 uses the discreteness of the set {Q(E) : E ∈ T} and Proposition 4.5 begins by taking Q to be a Q-coefficient quadratic form, but Definition 2.4 only provides an arbitrary real quadratic form. Please add one sentence explaining that a support form can be chosen with rational coefficients, for instance by adding a small rational positive definite form to a given support form; without this, the minimum argument in Lemma 4.6 is not justified as written.","section":"Lemma 4.6 and Proposition 4.5"},{"comment":"The assertion that for every θ ∈ [0, 1/2] the central charge of σ_{−t,s}[θ] lies in U_n^{>d} is stated without proof; a short computation or an explicit reference to the interlacing facts in Lemma C.13 and Lemma C.15 would clarify this load-bearing step in the equivalence of the two conjectures.","section":"Theorem 8.4, proof of Conjecture 8.1 implies Sbd, property (c)"},{"comment":"The jump from d(Pσ, Pσ′) = 1 to the existence of an infinite sequence of σ-stable objects whose σ-phases approach 1 and whose σ′-phases approach 0 deserves a one-sentence justification, since the equality case of the supremum in the definition of d does not by itself produce such a sequence.","section":"Proposition 2.16, proof of (3) ⇒ (2)"},{"comment":"The phrase 'U’y Proposition 2.12' appears to be a typo for 'By Proposition 2.12' in the proof of Corollary 2.14.","section":"Page 15, proof of Proposition 2.12"}],"recommendation":"minor_revision","confidential_remarks":"The structural half of the paper is strong and the conditional geometric applications are clearly labeled. I do not see grounds for rejection; the revision should focus on the clarifications listed in the minor comments, especially the rational-coefficient support form and the omitted computation in Theorem 8.4."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this is serious work and should be refereed. The reduced stability manifold Sb(T), the relation \\lesssim, and the reconstruction Stab(T) \\simeq TaSb(T) are genuinely new. The paper is also honest about where the main geometric theorem depends on a conjecture that may fail in some threefolds. The conditional scope is not a hidden flaw; it is flagged in the introduction and in Sections 7 and 8.\n\nWhat it does well: the equivalence relation on Stab(T) is cleanly defined, and Proposition 2.12 gives a solid local homeomorphism to real central charges. The wall-and-chamber preservation is a natural and useful consequence. The curve, surface, and threefold examples are assembled into one framework, which is valuable even where individual results were known. Proposition 6.4, the restriction theorem via \\lesssim, is a real improvement over earlier approaches, and its self-contained proof looks sound. The citation pattern is normal; prior work on curves, surfaces, and threefolds is acknowledged.\n\nThe soft spot is Proposition 4.5, which carries the reconstruction theorem. The proof is elaborate and depends on the quadratic form bookkeeping in Lemma 2.17 and on heart constancy along deformations. I went through the main steps and found no concrete failure; the stress test agrees. But this is the place where a referee should spend real time. If Lemma 2.17 had an edge case, the reconstruction would collapse. The argument is not machine-checked and is hard to visually verify, so I would not call the structural results beyond reasonable doubt at the first pass. That is not a reason to reject; it is a reason for careful refereeing.\n\nMinor issues: the paper is long and organization is somewhat uneven, with some technical lemmas deferred to appendices. The non-Hausdorff example on P1 is nice but a bit aside. I also found the notation Ta(\\tilde\\sigma) slightly overloaded, though not confusing once you read the definition.\n\nWho this is for: people working on Bridgeland stability, wall-crossing, and existence of stability conditions on higher-dimensional varieties. It deserves a serious referee and likely a visible slot. Send it out.","headline":"A genuinely new structural framework for Bridgeland stability, with an honest conditional geometric application; send it to a serious referee.","tokens_in":77158,"tokens_out":1538,"would_cite":true,"duration_ms":22032,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14F08","14K05","14J60","18G80"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper establishes a homeomorphism between the full stability manifold of Bridgeland stability conditions and a subspace of the tangent bundle of the reduced manifold determined by ≲.","keywords":["Derived category","Bridgeland stability conditions","reduced stability conditions","wall and chamber structure","restriction theorem","interlaced polynomials","stability manifold","moduli spaces"],"falsifier":"Find a rank-4 example of linear forms h, f1, f2 and a signature-(2,2) quadratic form Q for which the quadratic form Q̃ required by Lemma 2.17 provably does not exist; such an example would invalidate Proposition 4.5 and the reconstruction homeomorphism Stab(T) ≅ TaSb(T).","tokens_in":76135,"feed_emoji":"📐","tokens_out":8978,"duration_ms":89888,"temperature":0.7,"pith_summary":"Bridgeland stability conditions on a triangulated category form a complex manifold, but much of that manifold is redundant for wall-crossing. The paper introduces an equivalence relation that identifies stability conditions sharing the same imaginary central charge along connected fibers, and studies the quotient, called reduced stability conditions. The quotient is a real, possibly non-Hausdorff manifold of half the original dimension, and it retains the wall-and-chamber structure. The paper then defines a relation ≲ on the quotient and proves that the original stability manifold is homeomorphic to the subspace of the tangent bundle of the quotient selected by ≲. For smooth polarized varieties, the paper proposes a conjectural family of stability conditions whose reduced versions are parametrized by interlaced roots, and proves that if the conjecture holds, every smooth subvariety inherits stability conditions with geometric and vanishing properties.","feed_headline":"Reduced stability space rebuilds the full stability manifold","feed_subtitle":"Halving the stability manifold keeps its walls, and ≲ rebuilds the whole space.","key_machinery":"The load-bearing object is the reduced stability condition: an equivalence class of stability conditions under the relation ∼ that identifies conditions with equal imaginary central charge in the same path-connected fiber. Its companion is the relation ≲ on the quotient, defined by A_σ ⊂ P_τ(<1). The reconstruction is carried by Proposition 4.5, which characterizes the tangent space Ta(σ) of the reduced space purely through ≲, together with Lemma 2.17, a technical existence statement for quadratic forms of signature (2, ρ−2) that keep certain kernel unions inside a negativity region; these two ingredients produce the homeomorphism Stab(T) ≅ TaSb(T).","core_discovery":"On the paper's own terms, the central discovery is that the stability manifold Stab(T) is reducible to a smaller real manifold without losing information. Two stability conditions are declared equivalent when they have the same imaginary part of the central charge and lie in the same path-connected fiber of the imaginary-charge forgetful map; the quotient Sb(T) is a real, possibly non-Hausdorff manifold of half the dimension. The heart and the slice P(1) are well-defined on equivalence classes, and a second relation ≲, defined by inclusion of hearts, encodes the missing real part of the central charge. The main structural theorem (Corollary 4.8) is that π∼ × ForgReZ is a homeomorphism from Stab(T) to TaSb(T), the subspace of the tangent bundle of Sb(T) determined by ≲. On the geometric side, assuming Conjecture 1.1, the paper proves that the distinguished family Stab*_H(X) is unique up to even shifts, restricts to a stability condition on every smooth subvariety, makes skyscrapers stable, and satisfies a Bayer-type vanishing statement and an interlaced-root numerical bound.","pith_inferences":["The reconstruction theorem suggests that one could define stability conditions on a category by first constructing a reduced space Sb and a relation ≲, bypassing the harder problem of building the full stability manifold directly.","The interlaced-root parametrisation of reduced central charges invites a comparison with root systems and scattering diagrams; the paper already notes the P^2 example, and the threefold families may admit similar combinatorial descriptions.","The weakening parameter d in the Stab_d conjecture could serve as a quantitative measure of how far a variety is from admitting the strong family; testing where the threshold lies might organize threefolds by the failure of Bogomolov–Gieseker-type inequalities.","The non-Hausdorffness of Sb(T) is likely not a defect but the shadow of the gluing data: the explicit description on P^1 shows how degenerate points glue chambers, and understanding that pattern might explain how the full manifold is assembled from reduced data."],"forward_implications":["Wall-crossing for moduli spaces of stable objects can be studied on the half-dimensional reduced space, because the quotient map has convex fibers and preserves walls and chambers.","Any stability condition on a variety satisfying Conjecture 1.1 restricts to a stability condition on every smooth subvariety, so existence of stability conditions propagates from high dimension to low dimension.","The distinguished family Stab*_H(X), when it exists, is unique up to even homological shifts, making it a canonical slice of the stability manifold.","The numerical bound in Theorem 1.3.(5) says that the H-polarized character of a stable object is an alternating sum of gamma vectors with all coefficients of one sign, a statement that specializes to the Bogomolov inequality on surfaces and the Bogomolov–Gieseker-type inequalities on threefolds.","The weaker Stab_d conjecture, if true for large d, would give the same restriction and geometric conclusions for varieties where the strong conjecture fails."],"supporting_citations":[{"why":"Defines stability conditions and proves the local homeomorphism to central charges, giving Stab(T) its complex manifold structure.","marker":"[Bri07]"},{"why":"Supplies the effective deformation theorem and support-property formulation used in Proposition 2.16 and throughout the reconstruction.","marker":"[Bay19]"},{"why":"Provides the threefold stability conditions and the conjecture that implies Conjecture 1.1, plus the deformation lemma used to move central charges.","marker":"[BMS16]"},{"why":"Gives the restriction theorem (Corollary 2.2.2) that the paper's own restriction argument parallels and extends.","marker":"[Pol07]"},{"why":"Provides the support-property definition in the lattice form used to set up StabΛ(T).","marker":"[KS08]"}],"fun_headline_variants":["Stability manifold halves without losing walls","Half-dimension manifold rebuilds full stability space","Reduced stability space keeps walls, rebuilds all","Quotient stability space is half the dimension","Wall-chamber structure survives halved stability space"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that a technical quadratic-form existence statement with signature (2, ρ−2) holds, since it is what converts the local ≲-comparability of nearby reduced stability conditions into genuine tangent directions; without it the reconstruction homeomorphism collapses, and the geometric restriction theorem additionally assumes Conjecture 1.1, which the paper itself expects to fail for some threefolds.","fun_headline_variants_meta":{"raw":{"variants":["Stability manifold halves without losing walls","Half-dimension manifold rebuilds full stability space","Reduced stability space keeps walls, rebuilds all","Quotient stability space is half the dimension","Wall-chamber structure survives halved stability space"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00017,"raw_usage":{"total_tokens":1372,"prompt_tokens":1152,"completion_tokens":220,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":768,"completion_tokens_details":{"reasoning_tokens":150}},"tokens_in":768,"tokens_out":220,"duration_ms":2939,"temperature":1.0,"reasoning_tokens":150,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T22:13:37.579336+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a rank-4 example of linear forms h, f1, f2 and a signature-(2,2) quadratic form Q for which the quadratic form Q̃ required by Lemma 2.17 provably does not exist; such an example would invalidate Proposition 4.5 and the reconstruction homeomorphism Stab(T) ≅ TaSb(T).","supporting_citations":[],"review_version":1}