Stability conditions on some families of Calabi-Yau threefolds via orbifolding

Howard Nuer

REVIEW 2 major objections 4 minor 11 references

Stability conditions on some families of Calabi-Yau threefolds via orbifolding

T0 review · 2 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict The mirror quintic application is right, but Theorem 1 overreaches by assuming without proof that a G-invariant BMT stability condition exists. read the letter →

arxiv 2501.06207 v1 pith:W4XWJXQE submitted 2024-12-30 math.AG

Howard Nuer This is my paper · ORCID

classification math.AG MSC 14F0514J3218E30

keywords BridgelandstabilityconditionsCalabi-YauthreefoldsorbifoldingquinticmirrorequivariantderivedcategoriesMcKaycorrespondencesymmetry

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear Colleague,

The 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.

Where 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.

This 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.

The 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.

If 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.

Best.

Referee Report

2 major / 4 minor

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.

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 (2)

[Proof of Theorem 1] 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.
[Corollary 2] 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.

minor comments (4)

[Introduction] 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'.
[Theorem 1] 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].
[Proof of Theorem 1] 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.
[Corollary 2] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 1 is a conditional transfer via prior theorems; the unproved G-invariance choice is a gap, not a circular reduction.

full rationale

The derivation in Theorem 1 is a conditional transfer: it invokes the orbifolding hypothesis, the derived McKay equivalences [4,2], and the inducing theorem [8] to embed Stab(X)^G into Stab(Y), then chooses a stability condition from [1] in the invariant locus. No step uses the target statement as an input, no parameter is fitted and renamed as a prediction, and no definition is circular in terms of the conclusion. The references [1,8,4,2,6,11] are independent published results, and none are authored by the present paper's author, so there is no load-bearing self-citation. The only substantive concern is the unproved assertion that a BMT stability condition can be chosen with G-invariant (omega,B); this is a possible gap or missing hypothesis, not a circularity, because failure of that assertion would make the proof incomplete rather than make the conclusion equal to the assumptions. Corollary 2 applies Theorem 1 to Li's quintic stability conditions and Yu's 43 orbifold families; this is an external benchmark used as input, not a circular reuse of the conclusion. Score 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on four external theorems and one unproven assertion. The unproven assertion, the existence of a G-invariant stability condition on X, is the load-bearing gap. No free parameters or invented entities are introduced.

assumptions (5)

standard math D^b(Y) is equivalent to D^b_G(X) via the McKay correspondence and derived equivalence of crepant resolutions.
Invoked in the proof of Theorem 1 using [4, Theorem 1.2] and [2, Theorem 1.1].
standard math G-invariant stability conditions on X induce stability conditions on D^b_G(X).
From [8, Theorem 1.1].
domain assumption X admits Bridgeland stability conditions as constructed in [1], and for the quintic, [6].
This is the hypothesis of Theorem 1 and is provided by Li for the quintic.
domain assumption The orbifolding construction yields a crepant resolution Y with K_Y = f^* K_{X/G} = 0.
This is part of the definition of smooth orbifolding.
ad hoc to paper There exists a stability condition on X with G-invariant omega and B.
Asserted but not proven in the proof of Theorem 1. This is the main gap.

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Cite this review

Pith. "Pith review of Stability conditions on some families of Calabi-Yau threefolds via orbifolding." pith.science (2026). https://pith.science/paper/W4XWJXQE

@misc{pith2026250106207,
  author       = {Pith},
  title        = {Pith review of: Stability conditions on some families of Calabi-Yau threefolds via orbifolding},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W4XWJXQE}},
  note         = {Machine review of arXiv:2501.06207}
}

read the original abstract

We prove that families of Calabi-Yau threefolds (CY3's) admit Bridgeland stability conditions when they are obtained via orbifolding from a family of CY3's admitting Bridgeland stability conditions. In particular, we prove that the quintic mirror admits Bridgeland stability conditions.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

11 extracted references · 9 canonical work pages

[1]

Bayer, E

A. Bayer, E. Macr ` ı, and Y. Toda. Bridgeland stability co nditions on threefolds I: Bogomolov-Gieseker type in- equalities. J. Algebraic Geom. , 23(1):117–163, 2014. 1, 2

work page 2014
[9]

D. R. Morrison. Mirror symmetry and rational curves on qu intic threefolds: a guide for mathematicians. J. Amer. Math. Soc. , 6(1):223–247, 1993. 2

work page 1993
[11]

X. Yu. McKay correspondence and new Calabi-Yau threefo lds. Int. Math. Res. Not. IMRN , (21):6444–6468, 2017. 2 F aculty of Mathematics, Technion, Israel Institute of Tech nology Email address : hnuer@technion.ac.il

work page 2017
[2]

Bridgeland

T. Bridgeland. Flops and derived categories. Invent. Math. , 147(3):613–632, 2002. 1

work page 2002
[3]

Bridgeland

T. Bridgeland. Stability conditions on triangulated ca tegories. Ann. of Math. (2) , 166(2):317–345, 2007. 1

work page 2007
[4]

Bridgeland, A

T. Bridgeland, A. King, and M. Reid. The McKay correspond ence as an equivalence of derived categories. J. Amer. Math. Soc. , 14(3):535–554 (electronic), 2001. 1

work page 2001
[5]

N. Koseki. Stability conditions on Calabi-Yau double/t riple solids. arXiv:math/2007.00044, 2022. 1

arXiv 2007
[6]

C. Li. On stability conditions for the quintic threefold . Invent. Math. , 218(1):301–340, 2019. 1, 2

work page 2019

Show all 11 references

[7]

S. Liu. Stability condition on Calabi-Yau threefold of c omplete intersection of quadratic and quartic hypersurfac es. arXiv:math/2108.08934, 2022. 1

2022 arXiv
[8]

Macr ` ı, S

E. Macr ` ı, S. Mehrotra, and P. Stellari. Inducing stabil ity conditions. J. Algebraic Geom. , 18(4):605–649, 2009. 1

2009
[10]

Y. Toda. Gepner point and strong Bogomolov-Gieseker in equality for quintic 3-folds. In Higher dimensional alge- braic geometry—in honour of Professor Yujiro Kawamata’s si xtieth birthday , volume 74 of Adv. Stud. Pure Math. , pages 381–405. Math. Soc. Japan, Tokyo, 2017. 2

2017

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Bayer, E

A. Bayer, E. Macr ` ı, and Y. Toda. Bridgeland stability co nditions on threefolds I: Bogomolov-Gieseker type in- equalities. J. Algebraic Geom. , 23(1):117–163, 2014. 1, 2

work page 2014

[2] [9]

D. R. Morrison. Mirror symmetry and rational curves on qu intic threefolds: a guide for mathematicians. J. Amer. Math. Soc. , 6(1):223–247, 1993. 2

work page 1993

[3] [11]

X. Yu. McKay correspondence and new Calabi-Yau threefo lds. Int. Math. Res. Not. IMRN , (21):6444–6468, 2017. 2 F aculty of Mathematics, Technion, Israel Institute of Tech nology Email address : hnuer@technion.ac.il

work page 2017

[4] [2]

Bridgeland

T. Bridgeland. Flops and derived categories. Invent. Math. , 147(3):613–632, 2002. 1

work page 2002

[5] [3]

Bridgeland

T. Bridgeland. Stability conditions on triangulated ca tegories. Ann. of Math. (2) , 166(2):317–345, 2007. 1

work page 2007

[6] [4]

Bridgeland, A

T. Bridgeland, A. King, and M. Reid. The McKay correspond ence as an equivalence of derived categories. J. Amer. Math. Soc. , 14(3):535–554 (electronic), 2001. 1

work page 2001

[7] [5]

N. Koseki. Stability conditions on Calabi-Yau double/t riple solids. arXiv:math/2007.00044, 2022. 1

arXiv 2007

[8] [6]

C. Li. On stability conditions for the quintic threefold . Invent. Math. , 218(1):301–340, 2019. 1, 2

work page 2019