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arxiv: 2303.07095 · v3 · pith:NQJ4QQORnew · submitted 2023-03-13 · 🧮 math.AG

On the cone conjecture for Enriques manifolds

Gianluca Pacienza , Alessandra Sarti This is my paper

Pith reviewed 2026-05-24 10:02 UTC · model grok-4.3

classification 🧮 math.AG
keywords Enriques manifoldMorrison-Kawamata cone conjectureholomorphic symplectic manifoldmovable coneuniversal coverbirational geometryprime degree cover
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The pith

The Morrison-Kawamata cone conjecture holds for very general Enriques manifolds of prime cover degree.

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

The paper proves the Morrison-Kawamata cone conjecture for very general Enriques manifolds when the degree of their universal cover is prime. Enriques manifolds are non-simply connected manifolds whose covers are irreducible holomorphic symplectic manifolds, generalizing Enriques surfaces. The argument reduces the statement directly to the known case on the cover. A sympathetic reader would care because the conjecture asserts that the movable cone is polyhedral and generated by finitely many classes up to automorphisms, which organizes the birational geometry of these manifolds.

Core claim

We prove the Morrison-Kawamata cone conjecture for very general Enriques manifolds when the degree of the cover is prime. The proof uses the analogous result established by Amerik-Verbitsky for their universal cover. We also verify the conjecture for a very general Enriques manifold which is deformation equivalent to one of the known examples.

What carries the argument

The finite prime-degree étale cover from an irreducible holomorphic symplectic manifold to the Enriques manifold, which transfers the cone description from the cover to the quotient.

If this is right

  • The movable cone on such an Enriques manifold is rationally polyhedral.
  • Birational maps between these manifolds are determined by the action of the automorphism group on the cone.
  • The conjecture is confirmed for all known deformation classes of Enriques manifolds that admit a prime-degree cover.
  • The result gives a finite set of generators for the movable cone in these cases.

Where Pith is reading between the lines

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

  • The same reduction technique might work for composite cover degrees once the cover conjecture is known in those cases.
  • The approach suggests the cone conjecture extends to other finite quotients of holomorphic symplectic manifolds.
  • Removing the very general assumption would require separate analysis of special loci in the moduli space.

Load-bearing premise

The cone conjecture holds for the universal cover, and the Enriques manifold is very general in its deformation class.

What would settle it

An explicit very general Enriques manifold of prime cover degree whose movable cone requires infinitely many generators unrelated by automorphisms would disprove the claim.

read the original abstract

Enriques manifolds are non--simply connected manifolds whose universal cover is irreducible holomorphic symplectic, and as such they are natural generalizations of Enriques surfaces. The goal of this note is to prove the Morrison--Kawamata cone conjecture for very general Enriques manifolds when the degree of the cover is prime. The proof uses the analogous result (established by Amerik--Verbitsky) for their universal cover. We also verify the conjecture for a very general Enriques manifold which is deformation equivalent to one of the known examples.

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 / 1 minor

Summary. The manuscript proves the Morrison--Kawamata cone conjecture for very general Enriques manifolds whose universal cover has prime degree. The argument reduces the statement to the corresponding theorem of Amerik--Verbitsky for the irreducible holomorphic symplectic universal cover, via standard equivariant comparisons under a free prime-order action; it additionally verifies the conjecture explicitly for a very general Enriques manifold deformation-equivalent to one of the known examples.

Significance. If the reduction is valid, the result extends the cone conjecture to a natural class of non-simply-connected Calabi--Yau-type manifolds, generalizing the Enriques-surface and IHS cases. The explicit verification for a known deformation class supplies concrete evidence, and the reliance on the prior Amerik--Verbitsky theorem is a clear strength of the approach.

minor comments (1)
  1. [Abstract] Abstract: the phrase 'when the degree of the cover is prime' would benefit from an immediate parenthetical clarification that this means the order of the fundamental group (equivalently, the degree of the covering map) is a prime number.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive report and recommendation to accept the manuscript.

Circularity Check

0 steps flagged

No significant circularity; reduces to external theorem

full rationale

The paper's central derivation reduces the Morrison-Kawamata cone conjecture for very general Enriques manifolds (prime-degree cover) to the Amerik-Verbitsky theorem on the irreducible holomorphic symplectic universal cover. This is an external result by different authors. The quotient step uses standard equivariant cone comparisons under free prime-order action. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear. The argument is self-contained against the external benchmark and receives the default low score.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the external Amerik--Verbitsky theorem for the universal cover and on standard definitions of Enriques manifolds as quotients by free finite group actions on irreducible holomorphic symplectic manifolds; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption The Morrison--Kawamata cone conjecture holds for irreducible holomorphic symplectic manifolds (Amerik--Verbitsky).
    The paper reduces the Enriques case directly to this statement.

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

    math.AG 2025-04 unverdicted novelty 6.0

    Introduces well-clipped cones to prove the movable cone conjecture for finite quotients of Calabi-Yau type varieties and Galois descent for abelian varieties over perfect fields.

Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages · cited by 1 Pith paper

  1. [1]

    Amerik and M

    K. Amerik and M. Verbitsky, Morrison-Kawamata cone conjecture for hyperk\"ahler manifolds . Ann. Sci. \'Ec. Norm. Sup\'er.(4) 50 (2017), 973--993

    work page 2017

  2. [2]

    Amerik and M

    K. Amerik and M. Verbitsky, Collections of parabolic orbits in homogeneous spaces, homogeneous dynamics and hyperk\"ahler geometry . IMRN, Volume 2020, Issue 1, 25--38

    work page 2020

  3. [3]

    Beauville, Vari\'et\'es K\"ahleriennes dont la premi\`ere classe de Chern est nulle

    A. Beauville, Vari\'et\'es K\"ahleriennes dont la premi\`ere classe de Chern est nulle . J. Differential Geom. 18 (1983), no. 4, 755--782

    work page 1983

  4. [4]

    ahler manifolds with c_1= 0 . . Classification of algebraic and analytic manifolds, PM 39, 1--26; Birkh\

    A. Beauville, Some remarks on K\"ahler manifolds with c_1= 0 . . Classification of algebraic and analytic manifolds, PM 39, 1--26; Birkh\"auser (1983)

    work page 1983

  5. [5]

    Boissi\`ere, C

    S. Boissi\`ere, C. Camere and A. Sarti, Classification of automorphisms on a deformation family of hyperk\"ahler fourfolds by p-elementary lattices . Kyoto J. of Mathematics 56 (2016), no. 3, 465--499

    work page 2016

  6. [6]

    Boissi\`ere, M

    S. Boissi\`ere, M. Nieper-Wisskirchen and A. Sarti, Higher dimensional Enriques varieties and automorphisms of generalized Kummer varieties . J. Math. Pures et Appli. 95 (2011), 553--563

    work page 2011

  7. [7]

    Cascini, V

    P. Cascini, V. Lazi\'c, On the number of minimal models of a log smooth threefold , J. Math. Pures Appl. 102 (2014), 597--616

    work page 2014

  8. [8]

    I. V. Dolgachev and S. Kondo, Moduli of K3 surfaces and complex ball quotients, Arithmetic and geometry around hypergeometric functions, Progr. Math., vol. 260, Birkh\"auser, Basel, 2007, pp. 43--100

    work page 2007

  9. [9]

    Dolgachev, Mirror symmetry for lattice polarized K3 surfaces, J

    I.V. Dolgachev, Mirror symmetry for lattice polarized K3 surfaces, J. Math. Sci. 81 (1996), no. 3, 2599--2630

    work page 1996

  10. [10]

    Kawamata, On the cone of divisors of Calabi-Yau fiber spaces

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

    work page 1997

  11. [11]

    Fujiki On the de Rham cohomology group of a compact K\"ahler symplectic manifold

    A. Fujiki On the de Rham cohomology group of a compact K\"ahler symplectic manifold . Algebraic geometry, Sendai, 1985, volume 10 of Adv. Stud. Pure Math., p. 105--165. North-Holland, Amsterdam, 1987

    work page 1985

  12. [12]

    Gachet, H.-Y

    C. Gachet, H.-Y. Lin, L. Wang, Nef cones of fiber products and an application to the Cone Conjecture , preprint arXiv:2210.02779

    work page arXiv

  13. [13]

    Hassett, Yu

    B. Hassett, Yu. Tschinkel, Hodge theory and Lagrangian planes on generalized Kummer fourfolds. Moscow Mathematical Journal 13 (2013), no. 1, 33--56

    work page 2013

  14. [14]

    Horikawa, On deformations of holomorphic maps

    E. Horikawa, On deformations of holomorphic maps. III . Math. Ann. 222 (1976), no. 3., 275--282

    work page 1976

  15. [15]

    Huybrechts, Compact hyperk\"ahler manifolds: basic results

    D. Huybrechts, Compact hyperk\"ahler manifolds: basic results . Invent. Math. 135 (1999) 63--113

    work page 1999

  16. [16]

    Huybrechts, A global Torelli theorem for hyperk\"ahler manifolds [after M

    D. Huybrechts, A global Torelli theorem for hyperk\"ahler manifolds [after M. Verbitsky]. S\'eminaire Bourbaki: Vol. 2010/2011. Expos\'es 1027--1042. Ast\'erisque No. 348 (2012), Exp. No. 1040, x, 375--403

    work page 2010

  17. [17]

    Huybrechts, Lectures on K3 surfaces

    D. Huybrechts, Lectures on K3 surfaces . Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2016. doi:10.1017/CBO9781316594193

    work page doi:10.1017/cbo9781316594193 2016

  18. [18]

    Koll\'ar, Rational Curves on Algebraic Varieties

    J. Koll\'ar, Rational Curves on Algebraic Varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete 32, Springer Verlag, Berlin, 1996

    work page 1996

  19. [19]

    Lazi\'c, K

    V. Lazi\'c, K. Oguiso, Th. Peternell, The Morrison--Kawamata Cone Conjecture and Abundance on Ricci flat manifolds , Uniformization, Riemann-Hilbert Correspondence, Calabi-Yau manifolds and Picard-Fuchs Equations (L. Ji, S.-T. Yau, eds.), Advanced Lectures in Mathematics, vol. 42, International Press, 2018, pp. 157--185

    work page 2018

  20. [20]

    Ch. Lehn, G. Mongardi and G. Pacienza, The Morrison-Kawamata cone conjecture for singular symplectic varieties , preprint arXiv:2207.14754

    work page arXiv

  21. [21]

    Z. Li, H. Zhao On the relative Morrison-Kawamata cone conjecture , preprint arXiv:2206.13701v3

    work page arXiv

  22. [22]

    Markman, A survey of Torelli and monodromy results for holomorphic-symplectic varieties

    E. Markman, A survey of Torelli and monodromy results for holomorphic-symplectic varieties. Complex and differential geometry , 257--322, Springer Proc. Math., 8 , Springer, Heidelberg, 2011

    work page 2011

  23. [23]

    Markman and K

    E. Markman and K. Yoshioka A proof of the Kawamata-Morrison Cone Conjecture for holomorphic symplectic varieties of K3 or generalized Kummer deformation type . IMRN Volume 2015, Issue 24, 13563--13574

    work page 2015

  24. [24]

    Mongardi and M

    G. Mongardi and M. Wandel, Automorphisms of O'Grady's manifolds acting trivially on cohomology , Algebraic Geometry 4 (1) (2017) 104--119

    work page 2017

  25. [25]

    Morrison, Compactifications of moduli spaces inspired by mirror symmetry

    D. Morrison, Compactifications of moduli spaces inspired by mirror symmetry . Journ\'ees de g\'eom\'etrie alg\'ebrique d'Orsay (Orsay, 1992), Ast\'erisque 218 (1993), 243--271

    work page 1992

  26. [26]

    Namikawa, Periods of Enriques surfaces

    Y. Namikawa, Periods of Enriques surfaces . Math. Ann. 270 (1985), 201--222

    work page 1985

  27. [27]

    oer, Enriques manifolds . Journal f\

    K. Oguiso and S. Sch\"oer, Enriques manifolds . Journal f\"ur die reine und angewandte Mathematik, vol. 2011, no. 661, (2011), 215--235

    work page 2011

  28. [28]

    Oguiso and S

    K. Oguiso and S. Sch\"oer, Periods of Enriques manifolds . Pure and Applied Mathematics Quarterly Volume 7, Number 4 (Special Issue: In memory of Eckart Viehweg), (2011), 1631--1656

    work page 2011

  29. [29]

    Prendergast-Smith, The cone conjecture for abelian varieties

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

    work page 2012

  30. [30]

    Sterk, Finiteness results for algebraic K3 surfaces

    H. Sterk, Finiteness results for algebraic K3 surfaces . Math. Z. 189 (1985), 507--513

    work page 1985

  31. [31]

    On the finiteness of twists of irreduc ible symplectic varieties

    T. Takamatsu, On the finiteness of twists of irreducible symplectic varieties , preprint arXiv:2106.11651v2

    work page arXiv

  32. [32]

    Totaro, The cone conjecture for Calabi--Yau pairs in dimension two

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

    work page 2010

  33. [33]

    Totaro, Algebraic surfaces and hyperbolic geometry

    B. Totaro, Algebraic surfaces and hyperbolic geometry . Current Developments in Algebraic Geometry, ed. L. Caporaso, J. McKernan, M. Musta t a, and M. Popa, MSRI Publications 59 , Cambridge (2012), 405--426

    work page 2012

  34. [34]

    Verbitsky, A global Torelli theorem for hyperkahler manifolds, Duke Math

    M. Verbitsky, A global Torelli theorem for hyperkahler manifolds, Duke Math. J. 162 (15) (2013) 2929--2986

    work page 2013