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arxiv: 2309.02238 · v2 · submitted 2023-09-05 · 🧮 math.AG

The geometry of antisymplectic involutions, II

Laure Flapan , Emanuele Macr\`i , Kieran G. O'Grady , Giulia Sacc\`a This is my paper

Pith reviewed 2026-05-24 06:28 UTC · model grok-4.3

classification 🧮 math.AG
keywords antisymplectic involutionshyper-Kähler manifoldsK3^[n]-typefixed lociFano manifoldsBeauville-Bogomolov-Fujiki latticedivisibilitygeneral type varieties
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The pith

If the ample class has divisibility 2, one connected component of the fixed locus is a Fano manifold of index 3.

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

The paper examines antisymplectic involutions on projective hyper-Kähler manifolds of K3^[n]-type that arise from ample classes of square 2 in the Beauville-Bogomolov-Fujiki lattice. It proves that when this class has divisibility 2, one piece of the fixed set must be a Fano manifold of index 3. The result extends the known structure of the LLSvS eightfold tied to a cubic fourfold into higher dimensions. A second result identifies the remaining fixed component in that eightfold case as a variety of general type. These statements give concrete geometric descriptions of the fixed loci that such involutions produce.

Core claim

We prove that if the divisibility of the ample class is 2, then one connected component of the fixed locus is a Fano manifold of index 3, thus generalizing to higher dimensions the case of the LLSvS 8-fold associated to a cubic fourfold. We also show that, in the case of the LLSvS 8-fold associated to a cubic fourfold, the second component of the fixed locus is of general type, thus answering a question by Manfred Lehn.

What carries the argument

Antisymplectic involution induced by an ample class of square 2 and divisibility 2 in the Beauville-Bogomolov-Fujiki lattice on a projective hyper-Kähler manifold of K3^[n]-type.

If this is right

  • The fixed-locus description extends from the eight-dimensional LLSvS case to arbitrary n.
  • In the LLSvS eightfold the remaining fixed component is of general type.
  • The result resolves the open question on the type of the second fixed component in the cubic-fourfold case.

Where Pith is reading between the lines

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

  • The Fano component may supply new examples of Fano manifolds whose index is realized inside hyper-Kähler geometry.
  • The general-type component could constrain the possible Hodge numbers or birational models of the ambient manifold.
  • Direct computation of the fixed locus for small n greater than 2 would test whether the index-3 Fano property persists uniformly.

Load-bearing premise

The involution is an antisymplectic involution on a projective hyper-Kähler manifold of K3^[n]-type induced by an ample class of square 2 in the Beauville-Bogomolov-Fujiki lattice.

What would settle it

An explicit hyper-Kähler manifold of K3^[n]-type equipped with an antisymplectic involution from an ample class of square 2 and divisibility 2 whose fixed locus has no component that is a Fano manifold of index 3.

read the original abstract

We continue our study of fixed loci of antisymplectic involutions on projective hyper-K\"ahler manifolds of $\mathrm{K3}^{[n]}$-type induced by an ample class of square 2 in the Beauville-Bogomolov-Fujiki lattice. We prove that if the divisibility of the ample class is 2, then one connected component of the fixed locus is a Fano manifold of index 3, thus generalizing to higher dimensions the case of the LLSvS 8-fold associated to a cubic fourfold. We also show that, in the case of the LLSvS 8-fold associated to a cubic fourfold, the second component of the fixed locus is of general type, thus answering a question by Manfred Lehn.

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 / 2 minor

Summary. The manuscript continues the authors' study of fixed loci of antisymplectic involutions on projective hyper-Kähler manifolds of K3^[n]-type induced by an ample class of square 2 in the Beauville-Bogomolov-Fujiki lattice. The central claim is that when the divisibility of this ample class is 2, one connected component of the fixed locus is a Fano manifold of index 3, generalizing the LLSvS 8-fold case associated to a cubic fourfold. A second result shows that, in the LLSvS 8-fold case, the remaining component of the fixed locus is of general type, answering a question of Manfred Lehn.

Significance. If the stated results hold, the work supplies a higher-dimensional generalization of the geometry of fixed loci for antisymplectic involutions and resolves an open question on the second component in the cubic-fourfold setting. The claims are parameter-free in the sense that they follow from the given lattice-theoretic hypotheses without additional fitted constants, and they are falsifiable via explicit checks in low-dimensional cases such as the LLSvS 8-fold.

minor comments (2)
  1. [Introduction] The introduction would benefit from a brief reminder of the precise definition of divisibility for the ample class (currently referenced only via the BBF lattice).
  2. [§2] Notation for the two connected components of the fixed locus is introduced without an explicit label; a short sentence assigning symbols (e.g., X^+ and X^-) would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. No major comments were raised in the report.

Circularity Check

0 steps flagged

Minor self-citation of prior work but central claims remain independent

full rationale

The abstract and setup explicitly frame the work as a continuation of the authors' earlier study of antisymplectic involutions on K3^[n]-type hyper-Kähler manifolds. This constitutes a self-citation, but the load-bearing steps (divisibility-2 implying a Fano component of index 3, and general type for the second component in the cubic-fourfold case) are derived from external lattice-theoretic and Hodge-theoretic inputs rather than reducing to a fitted parameter or an unverified self-citation chain. No equation or theorem in the stated claims collapses by construction to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, axioms, or invented entities used in the proofs.

pith-pipeline@v0.9.0 · 5669 in / 1081 out tokens · 24536 ms · 2026-05-24T06:28:03.571863+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

47 extracted references · 47 canonical work pages

  1. [1]

    Alper, J., Good moduli spaces for Artin stacks, Ann. Inst. Fourier (Grenoble) 63 (2013), 2349--2402

    work page 2013

  2. [2]

    Alper, J., Halpern-Leistner, D., Heinloth, J., Existence of moduli spaces for algebraic stacks, eprint arXiv:1812.01128

    work page arXiv

  3. [3]

    Arbarello, E., Sacc\`a, G., Singularities of moduli spaces of sheaves on K3 surfaces and Nakajima quiver varieties, Adv. Math. 329 (2018), 649--703

    work page 2018

  4. [4]

    , Singularities for Bridgeland moduli spaces for K3 categories: an update, eprint arXiv:2307.07789

    work page arXiv

  5. [5]

    Bandiera, R., Manetti, M., Meazzini, F., Formality conjecture for minimal surfaces of Kodaira dimension 0, Compos. Math. 157 (2021), 215--235

    work page 2021

  6. [6]

    , Deformations of polystable sheaves on surfaces: quadraticity implies formality, Mosc. Math. J. 22 (2022), 239--263

    work page 2022

  7. [7]

    Bayer, A., Lahoz, M., Macr\`i, E., Nuer, H., Perry, A., Stellari, P., Stability conditions in families, Publ. Math. IH\'ES 133 (2021), 157--325

    work page 2021

  8. [8]

    Bayer, A., Macr\`i, E., MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations, Invent. Math. 198 (2014), 505--590

    work page 2014

  9. [9]

    Beckmann, T., Atomic objects on hyper-K\"ahler manifolds, eprint arXiv:2202.01184

    work page arXiv

  10. [10]

    Bridgeland, T., Stability conditions on triangulated categories. Ann. of Math. (2) 166 (2007), 317--345

    work page 2007

  11. [11]

    Budur, N., Zhang, Z., Formality conjecture for K3 surfaces, Compos. Math. 155 (2019), 902--911

    work page 2019

  12. [12]

    Camere, C., Cattaneo, A., Laterveer, R., On the Chow ring of certain Lehn--Lehn--Sorger--van Straten eightfolds, Glasg. Math. J. 64 (2022), 253--276

    work page 2022

  13. [13]

    Canonaco, A., Stellari, P., A tour about existence and uniqueness of dg enhancements and lifts, J. Geom. Phys. 122 (2017), 28--52

    work page 2017

  14. [14]

    , Uniqueness of dg enhancements for the derived category of a Grothendieck category, J. Eur. Math. Soc. (JEMS) 20 (2018), 2607--2641

    work page 2018

  15. [15]

    Castravet, A.-M., Mori dream spaces and blow-ups, in Algebraic geometry: Salt Lake City 2015, 143--167, Proc. Sympos. Pure Math. 97.1 , Amer. Math. Soc., Providence, RI, 2018

    work page 2015

  16. [16]

    Chen, H., Pertusi, L., Zhao, X., Deformation Theory and Formality Conjecture, preprint 2021

    work page 2021

  17. [17]

    126 (2001), 257--293

    Crawley-Boevey, W., Geometry of the moment map for representations of quivers, Compositio Math. 126 (2001), 257--293

    work page 2001

  18. [18]

    , Normality of Marsden--Weinstein reductions for representations of quivers, Math. Ann. 325 (2003), 55--79

    work page 2003

  19. [19]

    Debarre, O., Hyper-K\"ahler manifolds, Milan J. Math. 90 (2022), 305--387

    work page 2022

  20. [20]

    Dr\'ezet, J.-M., Le Potier, J., Fibr\'es stables et fibr\'es exceptionnels sur P^2 , Ann. Sci. \'Ecole Norm. Sup. (4) 18 (1985), 193--243

    work page 1985

  21. [21]

    Faltings, G., Some theorems about formal functions, Publ. Res. Inst. Math. Sci. 16 (1980), 721--737

    work page 1980

  22. [22]

    Ferretti, A., The Chow ring of double EPW sextics, Rend. Mat. Appl. (7) 31 (2011), 69--217

    work page 2011

  23. [23]

    Fiorenza, D., Iacono, D., Martinengo, E., Differential graded Lie algebras controlling infinitesimal deformations of coherent sheaves, J. Eur. Math. Soc. (JEMS) 14 (2012), 521--540

    work page 2012

  24. [24]

    Flapan, L., Macr\`i, E., O'Grady, K., Sacc\`a, G., The geometry of antysymplectic involutions, I, Math. Z. 300 (2022), 3457--3495

    work page 2022

  25. [25]

    Hassett, B., Tschinkel, Y., Intersection numbers of extremal rays on holomorphic symplectic varieties, Asian J. Math. 14 (2010), 303--322

    work page 2010

  26. [26]

    Huybrechts, D., Lehn, M., The geometry of moduli spaces of sheaves , Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2010

    work page 2010

  27. [27]

    King, A., Moduli of representations of finite-dimensional algebras, Quart. J. Math. Oxford Ser. (2) 45 (1994), 515--530

    work page 1994

  28. [28]

    Surveys Monogr

    Kleiman, S., The Picard scheme, in Fundamental algebraic geometry , 235--321, Math. Surveys Monogr. 123 , Amer. Math. Soc., Providence, RI, 2005

    work page 2005

  29. [29]

    Koll\'ar, J., Singularities of the minimal model program , Cambridge Tracts in Mathematics 200 , Cambridge University Press, Cambridge, 2013

    work page 2013

  30. [30]

    51/2015, 22--24, 2015

    Lehn, M., Twisted cubics on a cubic fourfold and in involution on the associated 8-dimensional symplectic manifold, in Oberwolfach Report No. 51/2015, 22--24, 2015

    work page 2015

  31. [31]

    Reine Angew

    Lehn, C., Lehn, M., Sorger, C., van Straten, D., Twisted cubics on cubic fourfolds, J. Reine Angew. Math. 731 (2017), 87--128

    work page 2017

  32. [32]

    15 (2006), 175--206

    Lieblich, M., Moduli of complexes on a proper morphism, J.\ Algebraic Geom. 15 (2006), 175--206

    work page 2006

  33. [33]

    Lunts, V., Orlov, D., Uniqueness of enhancement for triangulated categories, J. Amer. Math. Soc. 23 (2010), 853--908

    work page 2010

  34. [34]

    Notes Unione Mat

    Macr\`i, E., Schmidt, B., Lectures on Bridgeland stability, in Moduli of curves , 139--211, Lect. Notes Unione Mat. Ital. 21 , Springer, Cham, 2017

    work page 2017

  35. [35]

    Algebra 438 (2015), 90--118

    Manetti, M., On some formality criteria for DG-Lie algebras, J. Algebra 438 (2015), 90--118

    work page 2015

  36. [36]

    Reine Angew

    Markman, E., Modular Galois covers associated to symplectic resolutions of singularities, J. Reine Angew. Math. 644 (2010), 189--220

    work page 2010

  37. [37]

    Lecture Ser

    Nakajima, H., Lectures on Hilbert schemes of points on surfaces, Univ. Lecture Ser. 18 , American Mathematical Society, Providence, RI, 1999

    work page 1999

  38. [38]

    Namikawa, Y., Deformation theory of singular symplectic n-folds, Math. Ann. 319 (2001), 597--623

    work page 2001

  39. [39]

    Algebraic Geom

    O'Grady, K., The weight-two Hodge structure of moduli spaces of sheaves on a K3 surface, J. Algebraic Geom. , 6 (1997), 599--644

    work page 1997

  40. [40]

    , Irreducible symplectic 4-folds numerically equivalent to K3^ [2] , Commun. Contemp. Math. 10 (2008), 553--608

    work page 2008

  41. [41]

    , Covering families of Lagrangian subvarieties, preprint 2017

    work page 2017

  42. [42]

    Ohkawa, R., Moduli of Bridgeland semistable objects on P ^2 , Kodai Math. J. 33 (2010), 329--366

    work page 2010

  43. [43]

    Polishchuk, A., Van den Bergh, M., Semiorthogonal decompositions of the categories of equivariant coherent sheaves for some reflection groups, J. Eur. Math. Soc. (JEMS) 21 (2019), 2653--2749

    work page 2019

  44. [44]

    19 (1976), 306--381

    Procesi, C., The invariant theory of n n matrices, Advances in Math. 19 (1976), 306--381

    work page 1976

  45. [45]

    The Stacks Project Authors, The Stacks Project, 2023, available at http://stacks.math.columbia.edu

    work page 2023

  46. [46]

    Tajakka, T., Uhlenbeck Compactification as a Bridgeland Moduli Space, Int. Math. Res. Not. IMRN (2023), 4952--4997

    work page 2023

  47. [47]

    Yoshioka, K., Moduli spaces of stable sheaves on abelian surfaces, Math. Ann. 321 (2001), 817--884

    work page 2001