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arxiv: 2504.01753 · v4 · submitted 2025-04-02 · 🧮 math.AG

Well-clipped cones under finite quotients and applications to the cone conjecture

C\'ecile Gachet This is my paper

Pith reviewed 2026-05-22 21:59 UTC · model grok-4.3

classification 🧮 math.AG
keywords movable cone conjecturewell-clipped conesMorrison-Kawamata conjecturefinite quotientsCalabi-Yau varietiesabelian varietieshyperkähler manifoldsEnriques manifolds
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The pith

The movable cone conjecture holds for finite quotients of abelian varieties, hyperkähler manifolds, and related Calabi-Yau type varieties.

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

The paper defines a convexity property called well-clipped for cones in divisor spaces. Movable cones on abelian varieties and projective hyperkähler manifolds satisfy this property, which is preserved when passing to invariants under finite group actions and when taking direct sums. In this class the paper gives a criterion for the existence of a rational polyhedral fundamental domain under the natural group action. These facts are applied to prove the movable cone conjecture for finite quotients of products of such varieties, of smooth rational surfaces in klt Calabi-Yau pairs, and of Enriques manifolds; the same descent also yields the conjecture for abelian varieties over any perfect field.

Core claim

A convex cone is well-clipped when it satisfies a collection of combinatorial conditions that guarantee descent of the property under finite-group invariants and that permit a simple characterization of cones admitting a rational polyhedral fundamental domain. Movable cones of divisors on abelian varieties and projective hyperkähler manifolds are well-clipped; the property therefore descends to their finite quotients, establishing the movable cone conjecture in those cases and, by Galois descent, for abelian varieties over arbitrary perfect fields.

What carries the argument

The well-clipped property on convex cones, which encodes descent under finite group actions and supplies a criterion for rational polyhedral fundamental domains under the relevant automorphism group.

If this is right

  • The movable cone conjecture is true for finite quotients of products of projective primitive symplectic varieties.
  • The movable cone conjecture is true for finite quotients of abelian varieties.
  • The movable cone conjecture is true for Enriques manifolds.
  • Galois descent gives the movable Morrison-Kawamata cone conjecture for abelian varieties over any perfect field.

Where Pith is reading between the lines

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

  • The same descent argument may apply to other classes of varieties once their movable cones are shown to be well-clipped.
  • The framework separates the verification of the well-clipped property from the group-action descent step, allowing independent checks on each.
  • Over non-algebraically closed fields the Galois-descent statement reduces the conjecture to the complex case for abelian varieties.

Load-bearing premise

Movable cones on the source varieties are well-clipped and this property is preserved when passing to invariants under finite group actions.

What would settle it

A single movable cone on an abelian variety or hyperkähler manifold whose well-clipped property fails to descend to a finite quotient, or a finite quotient whose movable cone does not admit a rational polyhedral fundamental domain.

read the original abstract

We introduce a property of convex cones, being "well-clipped", that is inspired by the work of several complex algebraic geometers on the Morrison-Kawamata cone conjecture. That property is satisfied by movable cones of divisors on various complex projective varieties of Calabi-Yau type, such as abelian varieties and projective hyperk\"ahler manifolds. The property of being well-clipped has the advantage to descend under taking invariants by a finite group action, and to be stable under direct sums. In the class of well-clipped cones, we also provide a simple characterization of those cones that admit a rational polyhedral fundamental domain under some natural group action. We use this framework to prove the movable cone conjecture for finite quotients of various projective varieties of Calabi-Yau type, notably products of projective primitive symplectic varieties, abelian varieties, and smooth rational surfaces underlying klt Calabi-Yau pairs. This entails Enriques manifolds in the sense of Oguiso-Schr\"oer. We also provide Galois descent statements implying the movable Morrison-Kawamata cone conjecture for abelian varieties over arbitrary perfect fields.

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 paper introduces the notion of 'well-clipped' convex cones, inspired by work on the Morrison-Kawamata cone conjecture. It establishes that movable cones of divisors on abelian varieties and projective hyperkähler manifolds are well-clipped, proves that this property descends under invariants by finite group actions and is stable under direct sums, and gives a characterization within the class of well-clipped cones of those admitting a rational polyhedral fundamental domain under a natural group action. The framework is applied to prove the movable cone conjecture for finite quotients of products of projective primitive symplectic varieties, abelian varieties, and smooth rational surfaces underlying klt Calabi-Yau pairs (including Enriques manifolds in the sense of Oguiso-Schröer), as well as Galois descent statements for the movable Morrison-Kawamata cone conjecture on abelian varieties over arbitrary perfect fields.

Significance. If the results hold, the well-clipped property supplies a clean, descent-friendly framework that unifies and extends prior results on the cone conjecture to new classes of varieties and to quotients over arbitrary perfect fields. The stability under direct sums and the explicit characterization of cones with rational polyhedral fundamental domains are reusable tools. The concrete applications to Enriques manifolds and the Galois descent for abelian varieties constitute measurable progress in the birational geometry of Calabi-Yau-type varieties.

minor comments (3)
  1. [§1] §1 (Introduction): the statement that the property 'descends under taking invariants by a finite group action' would benefit from an explicit reference to the precise theorem number where descent is proved, rather than only to the abstract.
  2. [Definition 2.3] Definition 2.3: the definition of 'well-clipped' uses the phrase 'natural group action'; a short clarifying sentence on what 'natural' means in this context (e.g., the induced action on the Néron-Severi space) would improve readability for readers outside the immediate subfield.
  3. [Theorem 4.1] Theorem 4.1: the Galois descent statement for abelian varieties is stated over perfect fields; it would be helpful to record explicitly whether the proof uses only perfection or requires additional hypotheses that appear later in the argument.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and recommendation of minor revision. No major comments were listed in the report, so we have no specific points requiring point-by-point response. We will incorporate any minor suggestions in the revised manuscript.

Circularity Check

0 steps flagged

No significant circularity; new definition applied to independent base cases

full rationale

The paper introduces the new property 'well-clipped' for convex cones, verifies that movable cones on abelian varieties and projective hyperkähler manifolds satisfy it (citing prior external literature), proves descent under finite group invariants and stability under direct sums as theorems, and gives a characterization for rational polyhedral fundamental domains within this class. These are then applied to obtain the movable cone conjecture for the listed finite quotients and Galois descent statements. No equation or central claim reduces by construction to a fitted parameter, self-citation chain, or renamed input; the derivation chain is self-contained with independent content from the definition onward.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central contribution is the definition of the well-clipped property together with its stability and descent features; the work relies on standard facts from convex geometry and algebraic geometry rather than new free parameters or postulated entities.

axioms (1)
  • standard math Standard properties of convex cones in the Néron-Severi space and movable cone of divisors on projective varieties.
    Invoked when defining well-clipped cones and when discussing their behavior under group actions and direct sums.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. The Cone Conjecture for Enriques Surfaces in any Characteristic

    math.AG 2026-04 unverdicted novelty 7.0

    A characteristic-independent proof of the Morrison-Kawamata cone conjecture for Enriques surfaces based on generically finite degree-two morphisms.

Reference graph

Works this paper leans on

50 extracted references · 50 canonical work pages · cited by 1 Pith paper · 2 internal anchors

  1. [1]

    A. Ash, D. Mumford, M. Rapoport, and Y. Tai.Smooth compactification of locally symmetric varieties, volume Vol. IV ofLie Groups: History, Frontiers and Applications. Math Sci Press, Brookline, MA, 1975

    work page 1975

  2. [2]

    Bakker and C

    B. Bakker and C. Lehn. The global moduli theory of symplectic varieties.J. Reine Angew. Math., 790:223–265, 2022

    work page 2022

  3. [3]

    Beauville

    A. Beauville. Variétés kählériennes dont la première classe de Chern est nulle.J. Differ. Geom., 18:755–782, 1984

    work page 1984

  4. [4]

    Birkar, P

    C. Birkar, P. Cascini, C. D. Hacon, and J. McKernan. Existence of minimal models for varieties of log general type.J. Amer. Math. Soc., 23(2):405–468, 2010

    work page 2010

  5. [5]

    R. E. Borcherds. The Leech lattice and other lattices. arXiv:math/9911195

    work page internal anchor Pith review Pith/arXiv arXiv

  6. [6]

    Boucksom, J.-P

    S. Boucksom, J.-P. Demailly, M. Păun, and T. Peternell. The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension.J. Algebraic Geom., 22(2):201– 248, 2013

    work page 2013

  7. [7]

    Bright, A

    M. Bright, A. Logan, and R. van Luijk. Finiteness results for k3 surfaces over arbitrary fields. European Journal of Mathematics, 6(2):336–366, 2020

    work page 2020

  8. [8]

    Cantat and K

    S. Cantat and K. Oguiso. Birational automorphism groups and the movable cone theorem for Calabi-Yau manifolds of Wehler type via universal Coxeter groups. Amer. J. Math., 137(4):1013–1044, 2015

    work page 2015

  9. [9]

    O. Debarre. Un contre-exemple au théorème de Torelli pour les variétés symplectiques irré- ductibles. C. R. Acad. Sci. Paris Sér. I Math., 299(14):681–684, 1984

    work page 1984

  10. [10]

    St. Druel. A decomposition theorem for singular spaces with trivial canonical class of dimension at most five.Invent. Math., 211(1):245–296, 2018

    work page 2018

  11. [11]

    Faraut and A

    J. Faraut and A. Korányi.Analysis on symmetric cones. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 1994. Oxford Science Publications

    work page 1994

  12. [12]

    Gachet, H.-Y

    C. Gachet, H.-Y. Lin, Stenger I., and L. Wang. The effective cone conjecture for Calabi–Yau pairs. arXiv:2406.07307

    work page arXiv

  13. [13]

    Gachet, H.-Y

    C. Gachet, H.-Y. Lin, and L. Wang. Nef cones of fiber products and an application to the cone conjecture. Forum Math. Sigma, 12:Paper No. e28, 22, 2024

    work page 2024

  14. [14]

    Grassi and D

    A. Grassi and D. Morrison. Automorphisms and the Kähler cone of certain Calabi-Yau mani- folds. Duke Math. J., 71(3):831–838, 1993. 30 CÉCILE GACHET

    work page 1993

  15. [15]

    D. Greb, H. Guenancia, and S. Kebekus. Klt varieties with trivial canonical class: holonomy, differential forms, and fundamental groups.Geom. Topol., 23(4):2051–2124, 2019

    work page 2051

  16. [16]

    Hartshorne

    R. Hartshorne. Algebraic Geometry, volume 52 ofGraduate Texts in Mathematics. Springer, 1977

    work page 1977

  17. [17]

    A.HöringandT.Peternell.Algebraicintegrabilityoffoliationswithnumericallytrivialcanonical bundle. Invent. Math., 216(2):395–419, 2019

    work page 2019

  18. [18]

    Jordan, J

    P. Jordan, J. von Neumann, and E. Wigner. On an algebraic generalization of the quantum mechanical formalism.Ann. of Math. (2), 35(1):29–64, 1934

    work page 1934

  19. [19]

    Kapustka, G

    M. Kapustka, G. Mongardi, G. Pacienza, and P. Pokora. On the Boucksom–Zariski decomposi- tion for irreducible symplectic varieties and bounded negativity. arXiv:1911.03367

    work page arXiv 1911

  20. [20]

    Kawamata

    Y. Kawamata. On the cone of divisors of Calabi-Yau fiber spaces.Internat. J. Math., 8(5):665– 687, 1997

    work page 1997

  21. [21]

    Koecher.The Minnesota notes on Jordan algebras and their applications, volume 1710 of Lecture Notes in Mathematics

    M. Koecher.The Minnesota notes on Jordan algebras and their applications, volume 1710 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1999. Edited, annotated and with a preface by Aloys Krieg and Sebastian Walcher

    work page 1999

  22. [22]

    Kollár and S

    J. Kollár and S. Mori. Birational geometry of algebraic varieties, volume 134 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1998. With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original

    work page 1998

  23. [23]

    S. Lamy. The Cremona group. book project with a stable version available at:https://www. math.univ-toulouse.fr/~slamy/blog/cremona.html. July 2025

    work page 2025

  24. [24]

    J. M. Lee.Introduction to smooth manifolds, volume 218 ofGraduate Texts in Mathematics. Springer, New York, second edition, 2013

    work page 2013

  25. [25]

    C. Lehn, G. Mongardi, and G. Pacienza. The Morrison-Kawamata cone conjecture for singular symplectic varieties.Selecta Math. (N.S.), 30(4):Paper No. 79, 36, 2024

    work page 2024

  26. [26]

    On the relative Morrison-Kawamata cone conjecture

    Zhan Li and Hang Zhao. On the relative Morrison-Kawamata cone conjecture. arXiv:2206.13701v5

    work page arXiv

  27. [27]

    Looijenga

    E. Looijenga. Discrete automorphism groups of convex cones of finite type.Compos. Math., 150(11):1939–1962, 2014

    work page 1939

  28. [28]

    W. Lutz. The Morrison cone conjecture under deformation. arXiv:2410.05949v2

    work page arXiv

  29. [29]

    E. Markman. Integral constraints on the monodromy group of the hyperKähler resolution of a symmetric product of aK3 surface. Internat. J. Math., 21(2):169–223, 2010

    work page 2010

  30. [30]

    Math., pages 257–322

    E.Markman.AsurveyofTorelliandmonodromyresultsforholomorphic-symplecticvarieties.In Complex and differential geometry, volume 8 ofSpringer Proc. Math., pages 257–322. Springer, Heidelberg, 2011

    work page 2011

  31. [31]

    E. Markman. Prime exceptional divisors on holomorphic symplectic varieties and monodromy reflections. Kyoto J. Math., 53(2):345–403, 2013

    work page 2013

  32. [32]

    Markman and K

    E. Markman and K. Yoshioka. A proof of the Kawamata-Morrison cone conjecture for holomor- phic symplectic varieties of K3[n] or generalized Kummer deformation type.Int. Math. Res. Not. IMRN, (24):13563–13574, 2015

    work page 2015

  33. [33]

    Monti and A

    M. Monti and A. Quedo. The Kawamata–Morrison cone conjecture for generalized hyperelliptic variety. arXiv:2403.13156

    work page arXiv

  34. [34]

    D. R. Morrison. Beyond the Kähler cone. InProceedings of the Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993), volume 9 ofIsrael Math. Conf. Proc., pages 361–376. Bar-Ilan Univ., Ramat Gan, 1996

    work page 1993

  35. [35]

    On rational surfaces

    Masayoshi Nagata. On rational surfaces. II.Mem. Coll. Sci. Univ. Kyoto Ser. A. Math., 33:271– 293, 1960/61

    work page 1960

  36. [36]

    Namikawa

    Y. Namikawa. On the birational structure of certain Calabi-Yau threefolds.J. Math. Kyoto Univ., 31(1):151–164, 1991

    work page 1991

  37. [37]

    Namikawa

    Y. Namikawa. Counter-example to global Torelli problem for irreducible symplectic manifolds. Math. Ann., 324(4):841–845, 2002

    work page 2002

  38. [38]

    Engel, S

    Boundedness of some fibered K-trivial varieties. Engel, p. and filipazzi, s. and greer, f. and mauri, m. and svaldi, r. arXiv:2507.00973

    work page arXiv

  39. [39]

    Oguiso and J

    K. Oguiso and J. Sakurai. Calabi-Yau threefolds of quotient type.Asian J. Math., 5(1):43–77, 2001

    work page 2001

  40. [40]

    On the cone conjecture for Enriques manifolds

    G. Pacienza and A. Sarti. On the cone conjecture for enriques manifolds. arXiv:2303.07095v2

    work page internal anchor Pith review Pith/arXiv arXiv

  41. [41]

    Prendergast-Smith

    A. Prendergast-Smith. The cone conjecture for abelian varieties.J. Math. Sci. Univ. Tokyo, 19(2):243–261, 2012

    work page 2012

  42. [42]

    Yu. G. Prokhorov. Equivariant minimal model program.Uspekhi Mat. Nauk, 76(3(459)):93–182, 2021. ON THE CONE CONJECTURE FOR FINITE QUOTIENTS 31

    work page 2021

  43. [43]

    C. Schoen. On fiber products of rational elliptic surfaces with section.Math. Z., 197(2):177–199, 1988

    work page 1988

  44. [44]

    B. Skauli. The cone conjecture for some Calabi–Yau varieties. Master’s Thesis, University of Oslo, 2017, available athttps://www.duo.uio.no/bitstream/handle/10852/60868/Skauli_ Thesis.pdf?sequence=14

    work page 2017

  45. [45]

    Stenger and Z

    I. Stenger and Z. Xie. Cones of divisors on P3 blown up at eight very general points. arXiv:2303.12005v2

    work page arXiv

  46. [46]

    H. Sterk. Finiteness results for algebraicK3 surfaces. Math. Z., 189(4):507–513, 1985

    work page 1985

  47. [47]

    Suzuki.Group theory

    M. Suzuki.Group theory. I, volume 247 ofGrundlehren der Mathematischen Wissenschaften. Springer-Verlag, Berlin-New York, 1982. Translated from the Japanese by the author

    work page 1982

  48. [48]

    B. Totaro. The cone conjecture for Calabi-Yau pairs in dimension 2.Duke Math. J., 154(2):241– 263, 2010

    work page 2010

  49. [49]

    È.˜B. Vinberg. Discrete linear groups that are generated by reflections.Izv. Akad. Nauk SSSR Ser. Mat., 35:1072–1112, 1971

    work page 1971

  50. [50]

    F. Xu. On the cone conjecture for certain pairs of dimension at most 4. arXiv:2405.20899v2. Ruhr-Universität Bochum, Universitätsstr. 150, 44801 Bochum, Germany Email address: cecile.gachet@ruhr-uni-bochum.de

    work page arXiv