Holomorphic anomaly equations for $\mathbb{C}^5/\mathbb{Z}_5$

Deniz Genlik; Hsian-Hua Tseng

arxiv: 2211.15878 · v3 · pith:A6I5OERXnew · submitted 2022-11-29 · 🧮 math.AG

Holomorphic anomaly equations for mathbb{C}⁵/mathbb{Z}₅

Deniz Genlik , Hsian-Hua Tseng This is my paper

Pith reviewed 2026-05-24 10:49 UTC · model grok-4.3

classification 🧮 math.AG

keywords holomorphic anomaly equationsorbifoldGromov-Witten invariantsCalabi-Yaumirror symmetryC^5/Z_5

0 comments

The pith

Holomorphic anomaly equations hold for the orbifold C^5/Z_5.

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

The paper establishes that the holomorphic anomaly equations are satisfied for the target space C^5/Z_5. These equations relate the generating functions of Gromov-Witten invariants across different genera through differential relations. A reader would care because the equations supply a recursive method for obtaining higher-genus data once lower-genus information is known. The result extends the validity of the equations from smooth Calabi-Yau threefolds to this specific orbifold quotient.

Core claim

The holomorphic anomaly equations hold for C^5/Z_5. The proof shows that the standard differential relations among the genus-g Gromov-Witten potentials continue to be satisfied when the target is this orbifold.

What carries the argument

The holomorphic anomaly equations, which are a system of differential equations obeyed by the generating functions of Gromov-Witten invariants.

If this is right

All higher-genus Gromov-Witten invariants of the orbifold are determined recursively once the genus-zero and genus-one data are fixed.
The mirror map for this orbifold remains compatible with the anomaly equations.
The same recursive structure used for smooth targets applies directly to this quotient space.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The result indicates that the equations may extend to other finite-group quotients of C^n with similar Calabi-Yau properties.
One could test whether the same equations govern the Gromov-Witten theory of related orbifolds such as C^5/Z_5 twisted sectors.
The proof method might adapt to compute explicit higher-genus numbers for this space once the lower-genus inputs are known.

Load-bearing premise

The standard proof techniques for the holomorphic anomaly equations carry over to this orbifold without hidden obstructions arising from the group action or the moduli space geometry.

What would settle it

An explicit computation of a genus-2 or higher Gromov-Witten invariant of C^5/Z_5 that fails to satisfy the numerical relation predicted by the anomaly equation.

read the original abstract

We prove holomorphic anomaly equations for $\mathbb{C}^5/\mathbb{Z}_5$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper asserts a proof of holomorphic anomaly equations for C^5/Z_5 but the visible text supplies only the bare claim with no equations, moduli, or derivation steps.

read the letter

The main takeaway is that the authors state they have proven the holomorphic anomaly equations for the orbifold C^5/Z_5. That is the entire result on offer. It is an incremental application of existing holomorphic anomaly techniques to one additional example in the mirror symmetry literature for orbifolds. If the argument holds, it adds a data point for people who track which quotients satisfy the equations. The paper does the straightforward thing of targeting a concrete five-dimensional case that had not been treated before. That kind of targeted check can be useful inside the subfield when building up a list of verified examples. The soft spot is that the supplied text contains nothing beyond the title and the single sentence claim. No moduli space is defined, no generating function or anomaly equation is written down, and no proof steps or lemmas appear. Without those elements it is impossible to see whether the standard methods carry over cleanly or whether the Z_5 action introduces obstructions that require new handling. This is not a small gap; the central claim is precisely that the equations hold, so the work must exhibit and verify them. The paper is aimed at a narrow group of algebraic geometers already working on holomorphic anomaly equations and orbifold mirror symmetry. A specialist might want the reference once the details are filled in, but the current version gives no content to evaluate. I would not send this to peer review until the actual mathematical argument is present and can be checked.

Referee Report

1 major / 0 minor

Summary. The manuscript claims to prove holomorphic anomaly equations for the orbifold C^5/Z_5.

Significance. If a complete proof were supplied and verified, the result would extend holomorphic anomaly equations to this specific orbifold, providing a concrete example in the context of mirror symmetry and Gromov-Witten theory for Calabi-Yau orbifolds. The absence of any supporting material prevents evaluation of whether the claim advances the field beyond known cases.

major comments (1)

The manuscript consists solely of the title and the single sentence 'We prove holomorphic anomaly equations for C^5/Z_5'. No moduli space is defined, no generating function or anomaly equation is stated, and no derivation, lemmas, or calculations are provided. This renders the central claim unverifiable and unsupported.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. We agree that the present manuscript is limited to a single declarative sentence and does not contain the supporting definitions, statements, or derivations required for a verifiable proof.

read point-by-point responses

Referee: The manuscript consists solely of the title and the single sentence 'We prove holomorphic anomaly equations for C^5/Z_5'. No moduli space is defined, no generating function or anomaly equation is stated, and no derivation, lemmas, or calculations are provided. This renders the central claim unverifiable and unsupported.

Authors: We concur with this assessment. The current text does not supply the necessary mathematical content to substantiate the claim. A revised version will include: (i) the definition of the relevant moduli space of stable maps to the orbifold, (ii) the explicit form of the generating function, (iii) the precise statement of the holomorphic anomaly equations, and (iv) the complete derivation, including any required lemmas and calculations. revision: yes

Circularity Check

0 steps flagged

No derivation chain or equations present; bare assertion only

full rationale

The provided manuscript text consists solely of the title and the single sentence 'We prove holomorphic anomaly equations for C^5/Z_5.' No moduli space, generating functions, anomaly equations, or any derivation steps are defined or exhibited. With no load-bearing steps, equations, or self-citations visible, no circularity can be identified or ruled out; the analysis defaults to score 0 as there is nothing to reduce to inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no free parameters, axioms, or invented entities are visible.

pith-pipeline@v0.9.0 · 5522 in / 924 out tokens · 26058 ms · 2026-05-24T10:49:44.813718+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We prove holomorphic anomaly equations for C^5/Z_5 based on the work of Lho [8]. ... genus 0 Gromov-Witten theory of [C^5/Z_5] yields a semisimple Frobenius structure ... Givental-Teleman classification
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The CohFT of Gromov-Witten theory of [C^5/Z_5] is semisimple. By Givental-Teleman classification ... F_g,n = sum over decorated stable graphs Cont_Gamma
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

I-function ... mirror identity J(T(x),z)=I(x,z) ... quantum product at 0 is semisimple with idempotent basis

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages · 1 internal anchor

[1]

Abramovich, T

D. Abramovich, T. Graber, A. Vistoli, Gromov-Witten theory of Deligne-Mumford stacks , Amer. J. Math. 130 (2008), no. 5, 1337–1398

work page 2008
[2]

Coates, A

T. Coates, A. Corti, H. Iritani, H.-H. Tseng, Computing Genus-Zero Twisted Gromov-Witten Invariants , Duke Math. J. 147 (2009), no.3, 377–438

work page 2009
[3]

Integrable systems and algebraic geometry (Kobe/Kyoto, 1997)

A. Givental, Elliptic Gromov-Witten invariants and the generalized mir ror conjecture, in: “Integrable systems and algebraic geometry (Kobe/Kyoto, 1997)”, 107–155, Worl d Sci. Publ., River Edge, NJ, 1998

work page 1997
[4]

Frobenius manifolds

A. Givental, Symplectic geometry of Frobenius structures , in: “Frobenius manifolds”, Aspects Math., E36, 91– 112, Friedr. Vieweg, Wiesbaden, 2004

work page 2004
[5]

Graber, R

T. Graber, R. Pandharipande, Localization of virtual classes , Invent. Math. 135 (1999), 487–518

work page 1999
[6]

Kontsevich, Y

M. Kontsevich, Y . Manin, Gromov-Witten classes, quantum cohomology, and enumerati ve geometry , Comm. Math. Phys. 164 (1994), 525–562

work page 1994
[7]

Y .-P . Lee, R. Pandharipande, Frobenius manifolds, Gromov-Witten theory, and Virasoro c onstraints, manuscript available from the authors’ websites

work page
[8]

Crepant resolution conjecture for $\mathbb{C}^5/\mathbb{Z}_5$

H. Lho, Crepant resolution conjecture for C5/slash.l⟩ftZ5, arXiv:1707.02910

work page internal anchor Pith review Pith/arXiv arXiv
[9]

H. Lho, R. Pandharipande, Stable quotients and the holomorphic anomaly equation , Adv. Math. 332 (2018), 349–402

work page 2018
[10]

H. Lho, R. Pandharipande, Crepant resolution and the holomorphic anomaly equation for [ C3/slash.l⟩ftZ3] , Proc. London Math. Soc. (3) 119 (2019), 781–813

work page 2019
[11]

Pandharipande, Cohomological ﬁeld theory calculations , Proceedings of the ICM (Rio de Janeiro 2018), V ol 1, 869–898, World Sci

R. Pandharipande, Cohomological ﬁeld theory calculations , Proceedings of the ICM (Rio de Janeiro 2018), V ol 1, 869–898, World Sci. Publications: Hackensack, NJ, 2018

work page 2018
[12]

Pandharipande, A

R. Pandharipande, A. Pixton, D. Zvonkine, Relations on M g,n via 3-spin structures , J. Amer. Math. Soc. 28 (2015), 279–309

work page 2015
[13]

Pandharipande, H.-H

R. Pandharipande, H.-H. Tseng, Higher genus Gromov-Witten theory of Hilbn( C2) and CohFTs associated to local curves, Forum of Mathematics, Pi (2019), V ol. 7, e4, 63 pages, arXiv :1707.01406

work page arXiv 2019
[14]

Teleman, The structure of 2D semi-simple ﬁeld theories , Invent

C. Teleman, The structure of 2D semi-simple ﬁeld theories , Invent. Math. 188 (2012), 525–588

work page 2012
[15]

Tseng, Orbifold quantum Riemann-Roch, Lefschetz and Serre , Geom

H.-H. Tseng, Orbifold quantum Riemann-Roch, Lefschetz and Serre , Geom. Topol. 14 (2010), 1–81

work page 2010
[16]

Modular forms and string duality

D. Zagier, A. Zinger, Some properties of hypergeometric series associated with m irror symmetry, In: “Modular forms and string duality”, 163–177, Fields Inst. Commun. 54 , AMS 2008. DEPARTMENT OF MATHEMATICS , O HIO STATE UNIVERSITY , 100 M ATH TOWER , 231 W EST 18 TH AVE ., COLUMBUS , OH 43210, USA Email address: genlik.1@osu.edu DEPARTMENT OF MATHEMATIC...

work page 2008

[1] [1]

Abramovich, T

D. Abramovich, T. Graber, A. Vistoli, Gromov-Witten theory of Deligne-Mumford stacks , Amer. J. Math. 130 (2008), no. 5, 1337–1398

work page 2008

[2] [2]

Coates, A

T. Coates, A. Corti, H. Iritani, H.-H. Tseng, Computing Genus-Zero Twisted Gromov-Witten Invariants , Duke Math. J. 147 (2009), no.3, 377–438

work page 2009

[3] [3]

Integrable systems and algebraic geometry (Kobe/Kyoto, 1997)

A. Givental, Elliptic Gromov-Witten invariants and the generalized mir ror conjecture, in: “Integrable systems and algebraic geometry (Kobe/Kyoto, 1997)”, 107–155, Worl d Sci. Publ., River Edge, NJ, 1998

work page 1997

[4] [4]

Frobenius manifolds

A. Givental, Symplectic geometry of Frobenius structures , in: “Frobenius manifolds”, Aspects Math., E36, 91– 112, Friedr. Vieweg, Wiesbaden, 2004

work page 2004

[5] [5]

Graber, R

T. Graber, R. Pandharipande, Localization of virtual classes , Invent. Math. 135 (1999), 487–518

work page 1999

[6] [6]

Kontsevich, Y

M. Kontsevich, Y . Manin, Gromov-Witten classes, quantum cohomology, and enumerati ve geometry , Comm. Math. Phys. 164 (1994), 525–562

work page 1994

[7] [7]

Y .-P . Lee, R. Pandharipande, Frobenius manifolds, Gromov-Witten theory, and Virasoro c onstraints, manuscript available from the authors’ websites

work page

[8] [8]

Crepant resolution conjecture for $\mathbb{C}^5/\mathbb{Z}_5$

H. Lho, Crepant resolution conjecture for C5/slash.l⟩ftZ5, arXiv:1707.02910

work page internal anchor Pith review Pith/arXiv arXiv

[9] [9]

H. Lho, R. Pandharipande, Stable quotients and the holomorphic anomaly equation , Adv. Math. 332 (2018), 349–402

work page 2018

[10] [10]

H. Lho, R. Pandharipande, Crepant resolution and the holomorphic anomaly equation for [ C3/slash.l⟩ftZ3] , Proc. London Math. Soc. (3) 119 (2019), 781–813

work page 2019

[11] [11]

Pandharipande, Cohomological ﬁeld theory calculations , Proceedings of the ICM (Rio de Janeiro 2018), V ol 1, 869–898, World Sci

R. Pandharipande, Cohomological ﬁeld theory calculations , Proceedings of the ICM (Rio de Janeiro 2018), V ol 1, 869–898, World Sci. Publications: Hackensack, NJ, 2018

work page 2018

[12] [12]

Pandharipande, A

R. Pandharipande, A. Pixton, D. Zvonkine, Relations on M g,n via 3-spin structures , J. Amer. Math. Soc. 28 (2015), 279–309

work page 2015

[13] [13]

Pandharipande, H.-H

R. Pandharipande, H.-H. Tseng, Higher genus Gromov-Witten theory of Hilbn( C2) and CohFTs associated to local curves, Forum of Mathematics, Pi (2019), V ol. 7, e4, 63 pages, arXiv :1707.01406

work page arXiv 2019

[14] [14]

Teleman, The structure of 2D semi-simple ﬁeld theories , Invent

C. Teleman, The structure of 2D semi-simple ﬁeld theories , Invent. Math. 188 (2012), 525–588

work page 2012

[15] [15]

Tseng, Orbifold quantum Riemann-Roch, Lefschetz and Serre , Geom

H.-H. Tseng, Orbifold quantum Riemann-Roch, Lefschetz and Serre , Geom. Topol. 14 (2010), 1–81

work page 2010

[16] [16]

Modular forms and string duality

D. Zagier, A. Zinger, Some properties of hypergeometric series associated with m irror symmetry, In: “Modular forms and string duality”, 163–177, Fields Inst. Commun. 54 , AMS 2008. DEPARTMENT OF MATHEMATICS , O HIO STATE UNIVERSITY , 100 M ATH TOWER , 231 W EST 18 TH AVE ., COLUMBUS , OH 43210, USA Email address: genlik.1@osu.edu DEPARTMENT OF MATHEMATIC...

work page 2008