{"id":"8b61dc1a-980d-4ce4-8bd1-79447a80eb3e","arxiv_id":"2501.06207","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Calabi-Yau threefolds obtained via smooth orbifolding inherit Bridgeland stability conditions from the original threefold, including the mirror quintic.","lead":"This paper proves that any Calabi-Yau threefold obtained by orbifolding from another Calabi-Yau threefold that already has Bridgeland stability conditions also has them. That yields the first proof that the mirror quintic threefold admits Bridgeland stability conditions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1 assumes without proof that the BMT stability condition on X can be chosen G-invariant; for nontrivial G-actions on N^1(X) this is not automatic and no averaging argument is supplied.","rationale":"The Reader's weakest assumption is exactly the point that bites. I agree with the verdict: CONDITIONAL is the right status. The paper's overall strategy is credible: the derived McKay equivalence and the embedding of invariant stability conditions follow from standard theorems, and the application to the quintic mirror is plausible because the quintic has Picard number one so the G-action on the stability parameter space is trivial. The gap is localized in one line of the proof, but it is load-bearing for the stated generality of Theorem 1. The proof would be repaired by either citing a theorem that the set of BMT parameters can be chosen G-invariant, or by proving that the average of a BMT parameter remains in the existence region. Without that, the theorem overreaches its proof. I do not see an independent reason to reject the conclusion; the issue is evidential, not a known counterexample. Thus the verdict remains CONDITIONAL rather than ACCEPT or REJECT.","tokens_in":904,"tokens_out":856,"duration_ms":243323,"concrete_test":"Let sigma_{omega,B} in Stab(X) be any stability condition supplied by [1] and define (omega_bar, B_bar) = |G|^{-1} sum_{g in G} (g^*omega, g^*B). Recompute the BMT tilt construction of [1] for (omega_bar, B_bar): verify whether the numerical BG-type inequality in [1, Thm 1.1] holds and whether (A_{omega_bar,B_bar}, Z_{omega_bar,B_bar}) is a Bridgeland stability condition. If it is, the missing step is filled; if not, exhibit a concrete CY3 with a finite automorphism group acting nontrivially on N^1 (e.g., an abelian threefold quotient) where this computation fails. This single explicit check decides whether the invariance assumption in Theorem 1 is warranted.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central step in the proof of Theorem 1 is the sentence: 'Taking a stability condition sigma = (A_{omega,B}, Z_{omega,B}) on X as constructed in [1] with omega and B invariant under the G-action, we get sigma is a G-invariant stability condition.' This assertion is not proved. From the assumption that X admits some BMT stability condition it does not follow that one can choose a G-invariant one. If G acts nontrivially on N^1(X)_R, one can average classes to obtain G-invariant (omega^G,B^G), but the BMT construction [1] only produces a Bridgeland stability condition for parameters satisfying a non-linear numerical inequality; the averaged pair may lie outside that set. The paper gives no argument that the existence region is convex or closed under finite G-averages; the analogous region in Li's quintic theorem, alpha^2+(beta-floor(beta)-1/2)^2 > 1/4, is manifestly non-convex (it is the complement of a union of disks). Since [8, Thm 1.1] transfers only G-invariant stability conditions on X to Stab(D^b_G(X)) ~= Stab(Y), an empty Stab(X)^G would leave Stab(Y) without a construction. For the mirror quintic this gap is invisible because Pic(X) has rank 1, so every (omega,B) is G-invariant; for the general Theorem 1 the gap is real.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a criterion for Calabi-Yau threefolds (CY3s) obtained by smooth orbifolding to admit Bridgeland stability conditions. The main theorem states that if a smooth CY3 X admits Bridgeland stability conditions and Y is obtained from X by taking a finite group quotient followed by a crepant resolution, then Y admits Bridgeland stability conditions. The proof combines the derived McKay correspondence with an induction theorem for stability conditions under finite group actions, and the paper applies this to smooth quintic threefolds to conclude that the quintic mirror admits Bridgeland stability conditions.","tokens_in":3167,"tokens_out":5662,"duration_ms":59328,"significance":"If fully established, the result would be a welcome addition to the growing list of threefolds known to admit Bridgeland stability conditions, and the application to the quintic mirror would answer a notable open question. The paper is a short note that builds directly on the work of Bayer--Macrì--Toda, Li, Koseki, Liu, and Macrì--Mehrotra--Stellari; its proposed new contribution is the observation that orbifolding preserves the existence of stability conditions. The argument has a clear structure and cites the relevant literature, but as written it contains a load-bearing gap in the construction of a G-invariant stability condition, so the main theorem is not fully proved in the stated generality.","major_comments":[{"comment":"The proof asserts without proof that one can choose a stability condition sigma = (A_{omega,B}, Z_{omega,B}) on X with omega and B invariant under the G-action. This is not automatic when G acts nontrivially on N^1(X)_R. Averaging an arbitrary pair (omega,B) gives G-invariant classes (omega^G, B^G), but the BMT existence region is defined by non-linear inequalities such as those appearing in Corollary 2, alpha^2 + (beta - floor(beta) - 1/2)^2 > 1/4, which is not convex; the averaged parameters may fall outside the region. Since the induction theorem [8, Thm 1.1] requires Stab(X)^G to be nonempty, the proof of Theorem 1 is incomplete as stated. For the quintic mirror this gap is harmless because Pic(X) has rank one, so every (omega,B) is G-invariant, but the theorem is asserted for arbitrary smooth orbifoldings.","section":"Proof of Theorem 1"},{"comment":"The corollary claims that all 43 families of [11] obtained via smooth orbifolding from quintic CY3s admit stability conditions, but it does not verify that each family satisfies the hypotheses of Theorem 1, in particular that the group action can be chosen to fix the relevant polarization classes or that the BMT stability conditions can be made G-invariant. The qualifier 'essentially all possibilities' is also imprecise: either the list is covered or it is not. Please state the precise condition on the group action and check it for the listed families.","section":"Corollary 2"}],"minor_comments":[{"comment":"There are several typos: 'D-brains' should be 'D-branes', 'adm it' appears as a split word in the abstract, and 'In principal' in Remark 5 should be 'In principle'.","section":"Introduction"},{"comment":"The phrase 'as in [1]' is ambiguous: [1] constructs stability conditions under specific hypotheses (e.g., a Bogomolov-Gieseker inequality), so please spell out precisely what it means for X to admit stability conditions as in [1].","section":"Theorem 1"},{"comment":"The derivation of the equivalence D^b(Y) ≃ D^b_G(X) from [4, Theorem 1.2] and [2, Theorem 1.1] should state explicitly the hypotheses on the G-action and the crepant resolution (for example, G ⊂ SL(3) and Y a crepant resolution obtained from G-Hilb) so that the reader can check that they are satisfied in the intended orbifolding examples.","section":"Proof of Theorem 1"},{"comment":"The notation for the stability conditions sigma_{a,b}^{alpha,beta,H} is used without explanation; please define the parameters and the geometric meaning, especially since the paper does not reproduce Li's construction.","section":"Corollary 2"}],"recommendation":"major_revision","confidential_remarks":"The paper is a short note whose main theorem is a direct combination of known results once the G-invariant stability condition is available. The gap identified in the proof is real for the broad statement but is easily fixed in the quintic mirror application because Picard rank is one. I recommend major revision rather than rejection: the authors should either prove the existence of G-invariant stability conditions in the required generality or restrict the statement to cases where such invariance is automatic, and they should also tighten the claim about the 43 families in Corollary 2."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear Colleague,\n\nThe paper is a short note that does one thing: it points out that the BMT construction on a CY3, combined with the McKay equivalence and inducing stability, gives stability conditions on smooth resolutions of quotients by finite groups. The advertised application — the quintic mirror — is almost certainly correct. The proof there is clean because the quintic has Picard rank one, so every (ω,B) is invariant under the group action. That is a genuinely new data point for the field.\n\nWhere I have trouble is Theorem 1 as stated. The proof contains the sentence 'Taking a stability condition σ = (A_{ω,B}, Z_{ω,B}) on X as constructed in [1] with ω and B invariant under the G-action' without any argument that such a pair exists. The existence of some BMT stability condition on X does not imply the existence of one fixed by G. For a finite group acting nontrivially on N^1, the natural averaging of (ω,B) can land outside the admissible parameter region — the region in Li's quintic theorem is the complement of a union of disks, which is not convex. So the general theorem is not proven as stated.\n\nThis is not a fatal objection to the paper's main application. The mirror quintic construction in [9] uses a group of automorphisms coming from PGL(5), which acts trivially on the hyperplane class, so the invariant pair exists automatically. Corollary 2 is fine. But the theorem is stated far more broadly than the proof supports, and the gap should be acknowledged.\n\nThe note is well-written and honest about its scope (Remark 5 admits the general orbifolding method needs more work). The citation pattern is appropriate. There is no circularity and no invented entities.\n\nIf I were the editor, I'd send this to a referee. The main result, once narrowed to the invariant-polarization case or supplied with an averaging argument, is a nice addition to the literature. A referee should ask whether the G-invariant condition is automatic in the orbifolding families from [11], or whether Theorem 1 should be restated to include that hypothesis.\n\nBest.","headline":"The mirror quintic application is right, but Theorem 1 overreaches by assuming without proof that a G-invariant BMT stability condition exists.","tokens_in":3638,"tokens_out":7119,"would_cite":true,"duration_ms":72283,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14F05","14J32","18E30"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that smooth orbifold quotients of a Calabi-Yau threefold with Bridgeland stability conditions also admit them, and applies this to the quintic mirror.","keywords":["Bridgeland stability conditions","Calabi-Yau threefolds","orbifolding","quintic mirror","equivariant derived categories","derived McKay correspondence","mirror symmetry"],"falsifier":"Choose one of the 43 group actions on quintic threefolds and compute the fixed locus of the induced action on the stability parameter space $(\\alpha,\\beta,a,b)$ of the geometric construction; if the fixed locus is empty for some action, Theorem 1's proof does not apply to that orbifold family.","tokens_in":2681,"feed_emoji":"🪞","tokens_out":14192,"duration_ms":128201,"temperature":0.7,"pith_summary":"Bridgeland stability conditions turn the derived category of a variety into a geometric space, and constructing them on genuine Calabi-Yau threefolds is a central open problem. This note proves a transfer theorem: if a smooth Calabi-Yau threefold $X$ admits such stability conditions, then any smooth member $Y$ of a family obtained from $X$ by smooth orbifolding also admits them. Smooth orbifolding is the classical mirror-making recipe, dividing a family by a finite group and resolving the quotient crepantly, so the theorem bears directly on mirror families such as the quintic mirror. The proof runs through the derived McKay correspondence, which identifies the derived category of $Y$ with the equivariant derived category of $X$, and through the fact that $G$-invariant stability conditions on $X$ induce stability conditions on $Y$. Corollary 2 puts the quintic mirror on the list of Calabi-Yau threefolds that admit Bridgeland stability conditions.","feed_headline":"Orbifolding lifts stability conditions to the mirror quintic","feed_subtitle":"If a Calabi-Yau threefold has Bridgeland stability conditions, every smooth orbifold quotient inherits them.","key_machinery":"The load-bearing identity is the closed embedding $\\mathrm{Stab}(X)^G \\hookrightarrow \\mathrm{Stab}(D^b_G(X)) = \\mathrm{Stab}(Y)$ supplied by the inducing-stability theorem, together with the derived McKay correspondence $D^b(Y) \\cong D^b_G(X)$ for the crepant resolution $f: Y \\to X/G$. The correspondence carries objects of the equivariant derived category of $X$ to objects of the derived category of $Y$, and the embedding theorem ensures that the $G$-invariant locus of the stability manifold of $X$ is a submanifold of the stability manifold of $Y$. The only input needed is a stability condition on $X$ whose defining data $\\omega$ and $B$ are fixed by $G$; the mechanism then transcribes it to $Y$ with no further choices.","core_discovery":"The paper's central claim is Theorem 1: if $X$ is a smooth Calabi-Yau threefold admitting Bridgeland stability conditions of the kind constructed for threefolds in the paper's first reference, then every smooth member $Y$ of a family obtained from $X$ by smooth orbifolding admits Bridgeland stability conditions. Smooth orbifolding means that a finite group $G$ acts equivariantly on a subfamily, the quotient $X/G$ is resolved crepantly to $Y$, and the canonical classes satisfy $K_Y = f^{*} K_{X/G}$. The proof identifies derived categories via the McKay correspondence, $D^b(Y) \\cong D^b_G(X)$, and then applies the inducing theorem that embeds the $G$-invariant stability conditions on $X$ as a closed submanifold of the stability space of $Y$: $\\mathrm{Stab}(X)^G \\hookrightarrow \\mathrm{Stab}(D^b_G(X)) = \\mathrm{Stab}(Y)$. Since geometric stability conditions on quintic threefolds already exist, Corollary 2 concludes that all smooth members of the 43 orbifold families, quintic mirror included, carry stability conditions.","pith_inferences":["A concrete test of the proof is to compute, for each of the 43 group actions, whether the geometric stability parameters on the quintic have a nonempty $G$-invariant locus; the answer would delimit exactly where Theorem 1 applies.","The transfer is likely to extend beyond strict Calabi-Yau threefolds to any crepant resolution of a global quotient where the derived McKay correspondence and an inducing theorem are available.","If invariant stability conditions turn out to be rare, the orbifolding method could still be salvaged by using twisted or equivariant stability conditions whose equivariant structure replaces literal $G$-invariance.","An explicit description of $\\mathrm{Stab}(X)^G$ inside the stability manifold would give a concrete picture of part of the stringy Kähler moduli space of the mirror family, following the paper's Remark 4."],"forward_implications":["Every smooth Calabi-Yau threefold in the catalogue of 43 orbifold families built from quintics admits Bridgeland stability conditions.","The quintic mirror, the archetypal mirror-symmetric Calabi-Yau, admits Bridgeland stability conditions.","Any future existence theorem for stability conditions on a Calabi-Yau threefold automatically transfers to all smooth orbifold quotients of that threefold.","The same induction applies to the recently stabilized double and triple solids and $(2,4)$ complete intersections, so their smooth orbifold mirrors inherit stability conditions."],"supporting_citations":[{"why":"Supplies the construction of Bridgeland stability conditions on threefolds that Theorem 1 takes as input.","marker":"[1]"},{"why":"Together with [4], yields the derived-category equivalence $D^b(Y) \\cong D^b_G(X)$ for the crepant resolution.","marker":"[2]"},{"why":"Provides the derived McKay correspondence identifying the resolution's derived category with the equivariant derived category.","marker":"[4]"},{"why":"Constructs the geometric stability conditions on quintic threefolds that feed Corollary 2.","marker":"[6]"},{"why":"Gives the inducing theorem: $G$-invariant stability conditions on $X$ become stability conditions on $Y$.","marker":"[8]"},{"why":"Describes the quintic mirror as a smooth orbifolding of the quintic, placing it inside Corollary 2's scope.","marker":"[9]"},{"why":"Catalogues the 43 orbifold families from quintics that Corollary 2 covers.","marker":"[11]"}],"fun_headline_variants":["Orbifolding preserves Bridgeland stability on CY3s","Mirror quintic inherits stability via orbifolding","Smooth orbifolds keep Bridgeland stability conditions","New stability proof for Calabi-Yau orbifold quotients","Stability conditions lift from CY3 to its orbifold family"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes that on $X$ there is a stability condition whose defining data $\\omega$ and $B$ are fixed by the finite group $G$; the paper states this without proof, and if the group moves every such condition the transfer argument stops.","fun_headline_variants_meta":{"raw":{"variants":["Orbifolding preserves Bridgeland stability on CY3s","Mirror quintic inherits stability via orbifolding","Smooth orbifolds keep Bridgeland stability conditions","New stability proof for Calabi-Yau orbifold quotients","Stability conditions lift from CY3 to its orbifold family"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000656,"raw_usage":{"total_tokens":2951,"prompt_tokens":839,"completion_tokens":2112,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":455,"completion_tokens_details":{"reasoning_tokens":2025}},"tokens_in":455,"tokens_out":2112,"duration_ms":14957,"temperature":1.0,"reasoning_tokens":2025,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T23:08:21.678526+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Choose one of the 43 group actions on quintic threefolds and compute the fixed locus of the induced action on the stability parameter space $(\\alpha,\\beta,a,b)$ of the geometric construction; if the fixed locus is empty for some action, Theorem 1's proof does not apply to that orbifold family.","supporting_citations":[{"cited_title":"Bayer, E","cited_arxiv_id":null,"evidence_quote":"Supplies the construction of Bridgeland stability conditions on threefolds that Theorem 1 takes as input."},{"cited_title":"Bridgeland","cited_arxiv_id":null,"evidence_quote":"Together with [4], yields the derived-category equivalence $D^b(Y) \\cong D^b_G(X)$ for the crepant resolution."},{"cited_title":"Bridgeland, A","cited_arxiv_id":null,"evidence_quote":"Provides the derived McKay correspondence identifying the resolution's derived category with the equivariant derived category."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Constructs the geometric stability conditions on quintic threefolds that feed Corollary 2."},{"cited_title":"Macr ` ı, S","cited_arxiv_id":null,"evidence_quote":"Gives the inducing theorem: $G$-invariant stability conditions on $X$ become stability conditions on $Y$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Describes the quintic mirror as a smooth orbifolding of the quintic, placing it inside Corollary 2's scope."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Catalogues the 43 orbifold families from quintics that Corollary 2 covers."}],"review_version":1}