pith. sign in

arxiv: 2604.09962 · v1 · submitted 2026-04-11 · 🧮 math.AG

Descendant and Fourier-Mukai equivalences for simple flops

Jiun-Cheng Chen , Hsian-Hua Tseng This is my paper

Pith reviewed 2026-05-10 16:44 UTC · model grok-4.3

classification 🧮 math.AG
keywords simple flopsdescendant Gromov-Witten theoryFourier-Mukai equivalencebirational geometryderived categoriesgenus zero invariantsenumerative geometry
0
0 comments X

The pith

For a simple flop X dashrightarrow X', a correspondence equates genus-0 descendant Gromov-Witten theories compatibly with the induced Fourier-Mukai equivalence.

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

The paper constructs a correspondence linking the genus zero descendant Gromov-Witten theories of two varieties related by a simple flop. This correspondence equates the invariants computed on each side. It also establishes that this matching is compatible with the Fourier-Mukai equivalence of derived categories induced by the flop. Sympathetic readers would value this because it provides a bridge between quantum cohomology invariants and categorical equivalences in birational geometry.

Core claim

For a simple flop X dashrightarrow X', we construct a correspondence between genus 0 descendant Gromov-Witten theories of X and X'. We show that the Fourier-Mukai equivalence induced by X dashrightarrow X' is compatible, in a precise sense, with the descendant correspondence.

What carries the argument

The descendant correspondence for genus-0 Gromov-Witten theories induced by the simple flop, shown to be compatible with the associated Fourier-Mukai equivalence.

Load-bearing premise

The varieties are smooth projective over the complex numbers and admit a simple flop for which the standard definitions of Gromov-Witten theory and Fourier-Mukai equivalence apply.

What would settle it

An explicit calculation of the descendant Gromov-Witten invariants for a concrete simple flop example, such as the conifold, where the proposed correspondence fails to match the invariants.

read the original abstract

For a simple flop $X\dashrightarrow X'$, we construct a correspondence between genus $0$ descendant Gromov-Witten theories of $X$ and $X'$. We show that the Fourier-Mukai equivalence induced by $X\dashrightarrow X'$ is compatible, in a precise sense, with the descendant correspondence.

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.

Referee Report

0 major / 3 minor

Summary. The manuscript constructs an explicit correspondence between the genus-0 descendant Gromov-Witten theories of smooth projective varieties X and X' related by a simple flop X dashrightarrow X'. It further shows that the Fourier-Mukai equivalence induced by the flop is compatible with this correspondence, preserving descendant insertions and virtual classes, with the argument reducing to the local model of the exceptional P^1 with normal bundle O(-1) ⊕ O(-1).

Significance. If the result holds, it provides a concrete bridge between derived equivalences (via Fourier-Mukai) and enumerative geometry by extending ordinary GW invariance under simple flops to the descendant case. The explicit construction via direct comparison of generating functions after reduction to the local model, together with the verification that psi-class insertions and virtual classes are preserved, is a strength that makes the compatibility falsifiable and potentially useful for transferring invariants across birational transformations.

minor comments (3)
  1. The abstract states the existence of the correspondence and compatibility but provides no indication of the reduction to the local model or the direct comparison method used; expanding it slightly would improve accessibility without altering the technical content.
  2. In the introduction, the discussion of prior results on ordinary (non-descendant) GW invariance under flops would benefit from explicit citations to the relevant literature on the subject to better situate the descendant extension.
  3. Notation for the correspondence map between the theories (e.g., how it acts on the psi-classes and virtual classes) should be introduced with a single consistent symbol early in the text and used uniformly thereafter to avoid minor confusion in later sections.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our work and for recommending minor revision. No specific major comments were provided in the report, so we have no individual points to address. We are pleased that the referee recognizes the explicit construction and the reduction to the local model as strengths of the paper.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper constructs an explicit correspondence between genus-0 descendant Gromov-Witten theories for a simple flop X dashrightarrow X' and verifies compatibility with the induced Fourier-Mukai equivalence. This proceeds by reduction to the local model (P^1 with normal bundle O(-1)⊕O(-1)) using the standard definitions of stable map moduli spaces, virtual classes, descendant insertions, and FM kernels on derived categories. No steps reduce by construction to fitted parameters, self-definitions, or load-bearing self-citations; the central claims remain independent of the inputs and are verified by direct comparison of generating functions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard definitions of simple flops, genus-0 descendant Gromov-Witten theory, and Fourier-Mukai transforms; these are drawn from prior literature in algebraic geometry rather than introduced ad hoc here.

axioms (1)
  • standard math Standard properties of Gromov-Witten invariants and Fourier-Mukai equivalences for smooth projective varieties hold.
    Invoked implicitly when stating the correspondence and compatibility exist.

pith-pipeline@v0.9.0 · 5338 in / 1225 out tokens · 56322 ms · 2026-05-10T16:44:50.906777+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages

  1. [1]

    Semiorthogonal decomposition for algebraic varieties

    A. Bondal, D. Orlov,Semiorthogonal decomposition for algebraic varieties, arXiv:alg-geom/9506012

    work page Pith review arXiv

  2. [2]

    Chao, J.-C

    Q. Chao, J.-C. Chen, H.-H. Tseng,Crepant transformation correspondence for toric stack bundles, Adv. Geom. 26, No. 1, 105–125 (2026)

    work page 2026

  3. [3]

    Coates, A

    T. Coates, A. Givental,Quantum Riemann-Roch, Lefschetz and Serre, Ann. Math. (2) 165, No. 1, 15–53 (2007)

    work page 2007

  4. [4]

    Coates, A

    T. Coates, A. Givental, H.-H. Tseng,Virasoro constraints for toric bundles, Forum Math. Pi 12, Paper No. e4, 28 p. (2024)

    work page 2024

  5. [5]

    Coates, H

    T. Coates, H. Iritani, Y . Jiang,The crepant transformation conjecture for toric complete intersections, Adv. Math. 329, 1002–1087 (2018)

    work page 2018

  6. [6]

    Coates, H

    T. Coates, H. Iritani, H.-H. Tseng,Wall-crossings in toric Gromov-Witten theory. I: Crepant examples, Geom. Topol. 13, No. 5, 2675–2744 (2009)

    work page 2009

  7. [7]

    Coates, Y

    T. Coates, Y . Ruan,Quantum cohomology and crepant resolutions: a conjecture. (Cohomologie quantique et r´esolutions cr´epantes : une conjecture.), Ann. Inst. Fourier 63, No. 2, 431–478 (2013)

    work page 2013

  8. [8]

    Givental,Gromov-Witten invariants and quantization of quadratic Hamiltonians, Mosc

    A. Givental,Gromov-Witten invariants and quantization of quadratic Hamiltonians, Mosc. Math. J.4(2001), 551–568. 12 JIUN-CHENG CHEN AND HSIAN-HUA TSENG

    work page 2001

  9. [9]

    Frobenius manifolds. Quantum cohomology and singu- larities

    A. Givental,Symplectic geometry of Frobenius structures, in: “Frobenius manifolds. Quantum cohomology and singu- larities”, 91–112, Aspects of Mathematics E 36, Vieweg (2004)

    work page 2004

  10. [10]

    Iritani,An integral structure in quantum cohomology and mirror symmetry for toric orbifolds, Adv

    H. Iritani,An integral structure in quantum cohomology and mirror symmetry for toric orbifolds, Adv. Math. 222, No. 3, 1016–1079 (2009)

    work page 2009

  11. [11]

    Iwao, Y .-P

    Y . Iwao, Y .-P. Lee, H.-W. Lin, C.-L. Wang,Invariance of Gromov-Witten theory under a simple flop, J. Reine Angew. Math. 663, 67–90 (2012)

    work page 2012

  12. [12]

    Kawamata,D-equivalence and K-equivalence, J

    Y . Kawamata,D-equivalence and K-equivalence, J. Differ. Geom. 61, No. 1, 147–171 (2002)

    work page 2002

  13. [13]

    Koto,A mirror theorem for non-split toric bundles, Math

    Y . Koto,A mirror theorem for non-split toric bundles, Math. Ann. 393, No. 3-4, 3337–3394 (2025)

    work page 2025

  14. [14]

    Lee, H.-W

    Y .-P. Lee, H.-W. Lin, F. Qu, C.-L. Wang,Invariance of quantum rings under ordinary flops. III: A quantum splitting principle, Camb. J. Math. 4, No. 3, 333–401 (2016)

    work page 2016

  15. [15]

    Lee, H.-W

    Y .-P. Lee, H.-W. Lin, C.-L. Wang,Flops, motives, and invariance of quantum rings, Ann. Math. (2) 172, No. 1, 243–290 (2010)

    work page 2010

  16. [16]

    Lee, H.-W

    Y .-P. Lee, H.-W. Lin, C.-L. Wang,Invariance of quantum rings under ordinary flops. I: Quantum corrections and reduc- tion to local models, Algebr. Geom. 3, No. 5, 578–614 (2016)

    work page 2016

  17. [17]

    Lee, H.-W

    Y .-P. Lee, H.-W. Lin, C.-L. Wang,Invariance of quantum rings under ordinary flops. II: A quantum Leray-Hirsch theo- rem, Algebr. Geom. 3, No. 5, 615–653 (2016)

    work page 2016

  18. [18]

    Li,A degeneration formula of GW-invariants, J

    J. Li,A degeneration formula of GW-invariants, J. Differ. Geom. 60, No. 2, 199–293 (2002)

    work page 2002

  19. [19]

    J. Li, B. Wu,Good degeneration of Quot-schemes and coherent systems, Commun. Anal. Geom. 23, No. 4, 841–921 (2015)

    work page 2015

  20. [20]

    W. Lutz, Q. Shafi, R. Webb,Crepant Transformation Conjecture For the Grassmannian Flop, Trans. Am. Math. Soc. 378, No. 7, 4671–4706 (2025), arXiv:2404.12302

    work page arXiv 2025

  21. [21]

    Pandharipande, H.-H

    R. Pandharipande, H.-H. Tseng,TheHilb/Symcorrespondence forC 2: descendents and Fourier-Mukai, Math. Ann. 375(2019), 509–540

    work page 2019

  22. [22]

    Priddis, M

    N. Priddis, M. Shoemaker, Y . Wen,Wall crossing and the Fourier-Mukai transform for Grassmann flops, SIGMA 21 (2025), 008, 33 pages, arXiv:2404.12303

    work page arXiv 2025

  23. [23]

    Sankaran, V

    P. Sankaran, V . Uma,Cohomology of toric bundles, Comment. Math. Helv. 78, No. 3, 540–554 (2003); errata ibid. 79, 840–841 (2004)

    work page 2003

  24. [24]

    Second international congress of Chinese mathematicians. Pro- ceedings of the congress (ICCM2001), Taipei, Taiwan, December 17–22, 2001

    C.-L. Wang,K-equivalence in birational geometry, in: “Second international congress of Chinese mathematicians. Pro- ceedings of the congress (ICCM2001), Taipei, Taiwan, December 17–22, 2001”, New Studies in Advanced Mathematics 4, 199–216, International Press (2004). DEPARTMENT OFMATHEMATICS, THIRDGENERALBUILDING, NATIONALTSING-HUAUNIVERSITY, NO. 101 SEC....

    work page 2001