logo École polytechnique
logo Journal de l'École Polytechnique
The geometry of antisymplectic involutions, II
[La géométrie des involutions anti-symplectiques, II]
Journal de l’École polytechnique — Mathématiques, Tome 13 (2026), pp. 809-851

We continue our 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. 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.

Nous poursuivons notre étude des lieux fixes des involutions anti-symplectiques sur les variétés hyper-kählériennes projectives de type K3$^{[n]}$ induites par une classe ample de carré 2 dans le réseau de Beauville-Bogomolov-Fujiki. Nous prouvons que si la divisibilité de la classe ample est 2, alors une composante connexe du lieu fixe est une variété de Fano d’indice 3, généralisant ainsi aux dimensions supérieures le cas de la variété de LLSvS de dimension 8 associée à une variété cubique de dimension 4. Nous montrons également que, dans le cas de la variété de dimension 8 de LLSvS associée à une variété cubique de dimension 4, la deuxième composante du lieu fixe est de type général, répondant ainsi à une question posée par Manfred Lehn.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/jep.337
Classification : 14C20, 14D06, 14D20, 14F08, 14J42, 14J45, 14J60
Keywords: Projective hyper-Kähler manifolds, antisymplectic involutions, moduli spaces, Bridgeland stability, Fano manifolds, varieties of general type
Mots-clés : Variétés hyper-kählériennes projectives, involutions anti-symplectiques, espaces de modules, stabilité de Bridgeland, variétés de Fano, variétés de type général
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
Laure Flapan; Emanuele Macrì; Kieran G. O’Grady; Giulia Saccà. The geometry of antisymplectic involutions, II. Journal de l’École polytechnique — Mathématiques, Tome 13 (2026), pp. 809-851. doi: 10.5802/jep.337
@article{JEP_2026__13__809_0,
     author = {Laure Flapan and Emanuele Macr{\`\i} and Kieran G. O{\textquoteright}Grady and Giulia Sacc\`a},
     title = {The geometry of antisymplectic involutions, {II}},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {809--851},
     year = {2026},
     publisher = {\'Ecole polytechnique},
     volume = {13},
     doi = {10.5802/jep.337},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.337/}
}
TY  - JOUR
AU  - Laure Flapan
AU  - Emanuele Macrì
AU  - Kieran G. O’Grady
AU  - Giulia Saccà
TI  - The geometry of antisymplectic involutions, II
JO  - Journal de l’École polytechnique — Mathématiques
PY  - 2026
SP  - 809
EP  - 851
VL  - 13
PB  - École polytechnique
UR  - https://jep.centre-mersenne.org/articles/10.5802/jep.337/
DO  - 10.5802/jep.337
LA  - en
ID  - JEP_2026__13__809_0
ER  - 
%0 Journal Article
%A Laure Flapan
%A Emanuele Macrì
%A Kieran G. O’Grady
%A Giulia Saccà
%T The geometry of antisymplectic involutions, II
%J Journal de l’École polytechnique — Mathématiques
%D 2026
%P 809-851
%V 13
%I École polytechnique
%U https://jep.centre-mersenne.org/articles/10.5802/jep.337/
%R 10.5802/jep.337
%G en
%F JEP_2026__13__809_0

[AHLH23] J. Alper, D. Halpern-Leistner & J. Heinloth - “Existence of moduli spaces for algebraic stacks”, Invent. Math. 234 (2023) no. 3, p. 949-1038 | Zbl | DOI | MR

[Alp13] J. Alper - “Good moduli spaces for Artin stacks”, Ann. Inst. Fourier (Grenoble) 63 (2013) no. 6, p. 2349-2402 | DOI | Numdam | MR | Zbl

[AS18] E. Arbarello & G. Saccà - “Singularities of moduli spaces of sheaves on K3 surfaces and Nakajima quiver varieties”, Adv. Math. 329 (2018), p. 649-703 | Zbl | DOI | MR

[AS25] E. Arbarello & G. Saccà - “Singularities of Bridgeland moduli spaces for K3 categories: an update”, in Perspectives on four decades of algebraic geometry. Vol. 1. In memory of Alberto Collino, Progress in Math., vol. 351, Birkhäuser/Springer, Cham, 2025, p. 1-42 | DOI | Zbl | MR

[BD85] A. Beauville & R. Donagi - “La variété des droites d’une hypersurface cubique de dimension 4”, C. R. Acad. Sci. Paris Sér. I Math. 301 (1985) no. 14, p. 703-706 | Zbl | MR

[Bec25] T. Beckmann - “Atomic objects on hyper-Kähler manifolds”, J. Algebraic Geom. 34 (2025) no. 1, p. 109-160 | DOI | Zbl | MR

[BLM + 21] A. Bayer, M. Lahoz, E. Macrì, H. Nuer, A. Perry & P. Stellari - “Stability conditions in families”, Publ. Math. Inst. Hautes Études Sci. 133 (2021), p. 157-325 | DOI | Numdam | Zbl | MR

[BM14] A. Bayer & E. Macrì - “MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations”, Invent. Math. 198 (2014) no. 3, p. 505-590 | DOI | Zbl | MR

[BMM21] R. Bandiera, M. Manetti & F. Meazzini - “Formality conjecture for minimal surfaces of Kodaira dimension 0”, Compositio Math. 157 (2021) no. 2, p. 215-235 | DOI | Zbl | MR

[BMM22] R. Bandiera, M. Manetti & F. Meazzini - “Deformations of polystable sheaves on surfaces: quadraticity implies formality”, Moscow Math. J. 22 (2022) no. 2, p. 239-263 | DOI | Zbl | MR

[Bri07] T. Bridgeland - “Stability conditions on triangulated categories”, Ann. of Math. (2) 166 (2007) no. 2, p. 317-345 | DOI | Zbl | MR

[BZ19] N. Budur & Z. Zhang - “Formality conjecture for K3 surfaces”, Compositio Math. 155 (2019) no. 5, p. 902-911 | DOI | Zbl | MR

[Cas18] A.-M. Castravet - “Mori dream spaces and blow-ups”, in Algebraic geometry (Salt Lake City, 2015), Proc. Sympos. Pure Math., vol. 97.1, American Mathematical Society, Providence, RI, 2018, p. 143-167 | DOI

[CB01] W. Crawley-Boevey - “Geometry of the moment map for representations of quivers”, Compositio Math. 126 (2001) no. 3, p. 257-293 | DOI | Zbl | MR

[CB03] W. Crawley-Boevey - “Normality of Marsden-Weinstein reductions for representations of quivers”, Math. Ann. 325 (2003) no. 1, p. 55-79 | DOI | Zbl | MR

[CCL22] C. Camere, A. Cattaneo & R. Laterveer - “On the Chow ring of certain Lehn–Lehn–Sorger–van Straten eightfolds”, Glasgow Math. J. 64 (2022) no. 2, p. 253-276 | DOI | Zbl | MR

[CPZ24] H. Chen, L. Pertusi & X. Zhao - “Some remarks about deformation theory and formality conjecture”, Ann. Univ. Ferrara Sez. VII 70 (2024) no. 3, p. 761-779 | DOI | Zbl

[CS17] A. Canonaco & P. Stellari - “A tour about existence and uniqueness of dg enhancements and lifts”, J. Geom. Phys. 122 (2017), p. 28-52 | DOI | MR | Zbl

[CS18] A. Canonaco & P. Stellari - “Uniqueness of dg enhancements for the derived category of a Grothendieck category”, J. Eur. Math. Soc. (JEMS) 20 (2018) no. 11, p. 2607-2641 | DOI | MR | Zbl

[Deb22] O. Debarre - “Hyper-Kähler manifolds”, Milan J. Math. 90 (2022) no. 2, p. 305-387 | DOI | MR | Zbl

[DK18] O. Debarre & A. Kuznetsov - “Gushel-Mukai varieties: classification and birationalities”, Algebraic Geom. 5 (2018) no. 1, p. 15-76 | DOI | MR | Zbl

[DLP85] J.-M. Drezet & J. Le Potier - “Fibrés stables et fibrés exceptionnels sur P 2 ”, Ann. Sci. École Norm. Sup. (4) 18 (1985) no. 2, p. 193-243 | DOI | MR | Zbl

[DV10] O. Debarre & C. Voisin - “Hyper-Kähler fourfolds and Grassmann geometry”, J. reine angew. Math. 649 (2010), p. 63-87 | DOI | MR | Zbl

[Fal80] G. Faltings - “Some theorems about formal functions”, Publ. RIMS, Kyoto Univ. 16 (1980) no. 3, p. 721-737 | DOI | Zbl

[Fer11] A. Ferretti - “The Chow ring of double EPW sextics”, Rend. Mat. (7) 31 (2011) no. 3-4, p. 69-217 | MR | Zbl

[FIM12] D. Fiorenza, D. Iacono & E. Martinengo - “Differential graded Lie algebras controlling infinitesimal deformations of coherent sheaves”, J. Eur. Math. Soc. (JEMS) 14 (2012) no. 2, p. 521-540 | DOI | MR | Zbl

[FM21] E. Fatighenti & G. Mongardi - “Fano varieties of K3-type and IHS manifolds”, Internat. Math. Res. Notices (2021) no. 4, p. 3097-3142 | DOI | Zbl | MR

[FMOS22] L. Flapan, E. Macrì, K. G. O’Grady & G. Saccà - “The geometry of antisymplectic involutions, I”, Math. Z. 300 (2022) no. 4, p. 3457-3495 | DOI | MR | Zbl

[HL10] D. Huybrechts & M. Lehn - The geometry of moduli spaces of sheaves, Cambridge Math. Library, Cambridge University Press, Cambridge, 2010 | DOI | MR | Zbl

[HT10] B. Hassett & Y. Tschinkel - “Intersection numbers of extremal rays on holomorphic symplectic varieties”, Asian J. Math. 14 (2010) no. 3, p. 303-322 | DOI | MR | Zbl

[IM15] A. Iliev & L. Manivel - “Fano manifolds of Calabi-Yau Hodge type”, J. Pure Appl. Algebra 219 (2015) no. 6, p. 2225-2244 | DOI | MR | Zbl

[IM19] A. Iliev & L. Manivel - “Hyperkähler manifolds from the Tits-Freudenthal magic square”, European J. Math. 5 (2019) no. 4, p. 1139-1155 | MR | Zbl | DOI

[Kin94] A. D. King - “Moduli of representations of finite-dimensional algebras”, Quart. J. Math. Oxford Ser. (2) 45 (1994) no. 180, p. 515-530 | DOI | MR | Zbl

[Kle05] S. L. Kleiman - “The Picard scheme”, in Fundamental algebraic geometry, Math. Surveys Monogr., vol. 123, American Mathematical Society, Providence, RI, 2005, p. 235-321

[Kol13] J. Kollár - Singularities of the minimal model program, Cambridge Tracts in Math., vol. 200, Cambridge University Press, Cambridge, 2013 | DOI | MR | Zbl

[Kuh61] N. Kuhlmann - “Die Normalisierung komplexer Räume”, Math. Ann. 144 (1961), p. 110-125 | DOI | MR | Zbl

[Leh16] M. Lehn - “Twisted cubics on a cubic fourfold and in involution on the associated 8-dimensional symplectic manifold”, in Mini-workshop: singular curves on K3 surfaces and hyperkähler manifolds, Oberwolfach Report, vol. 12, no. 4, European Mathematical Society, 2016, p. 2962-2964 | DOI

[Li93] J. Li - “Algebraic geometric interpretation of Donaldson’s polynomial invariants”, J. Differential Geom. 37 (1993) no. 2, p. 417-466 | DOI | MR | Zbl

[Lie06] M. Lieblich - “Moduli of complexes on a proper morphism”, J. Algebraic Geom. 15 (2006) no. 1, p. 175-206 | DOI | MR | Zbl

[LLSvS17] C. Lehn, M. Lehn, C. Sorger & D. van Straten - “Twisted cubics on cubic fourfolds”, J. reine angew. Math. 731 (2017), p. 87-128 | DOI | MR | Zbl

[LO10] V. A. Lunts & D. O. Orlov - “Uniqueness of enhancement for triangulated categories”, J. Amer. Math. Soc. 23 (2010) no. 3, p. 853-908 | DOI | MR | Zbl

[LP92] J. Le Potier - “Fibré déterminant et courbes de saut sur les surfaces algébriques”, in Complex projective geometry (Trieste, 1989/Bergen, 1989), London Math. Soc. Lecture Note Ser., vol. 179, Cambridge University Press, Cambridge, 1992, p. 213-240 | DOI | Zbl

[LPZ23] C. Li, L. Pertusi & X. Zhao - “Twisted cubics on cubic fourfolds and stability conditions”, Algebraic Geom. 10 (2023) no. 5, p. 620-642 | DOI | MR | Zbl

[Man15] M. Manetti - “On some formality criteria for DG-Lie algebras”, J. Algebra 438 (2015), p. 90-118 | DOI | MR | Zbl

[Mar10] E. Markman - “Modular Galois covers associated to symplectic resolutions of singularities”, J. reine angew. Math. 644 (2010), p. 189-220 | DOI | MR | Zbl

[MS17] E. Macrì & B. Schmidt - “Lectures on Bridgeland stability”, in Moduli of curves, Lect. Notes Unione Mat. Ital., vol. 21, Springer, Cham, 2017, p. 139-211 | DOI | Zbl

[Muk87] S. Mukai - “On the moduli space of bundles on K3 surfaces. I”, in Vector bundles on algebraic varieties (Bombay, 1984), Tata Inst. Fund. Res. Stud. Math., vol. 11, Tata Institure of Fundamental Research, Bombay, 1987, p. 341-413 | Zbl

[Nak99] H. Nakajima - Lectures on Hilbert schemes of points on surfaces, University Lect. Series, vol. 18, American Mathematical Society, Providence, RI, 1999 | DOI | MR | Zbl

[Nam01] Y. Namikawa - “Deformation theory of singular symplectic n-folds”, Math. Ann. 319 (2001) no. 3, p. 597-623 | DOI | MR | Zbl

[Ohk10] R. Ohkawa - “Moduli of Bridgeland semistable objects on P 2 ”, Kodai Math. J. 33 (2010) no. 2, p. 329-366 | DOI | MR | Zbl

[O’G97] K. G. O’Grady - “The weight-two Hodge structure of moduli spaces of sheaves on a K3 surface”, J. Algebraic Geom. 6 (1997) no. 4, p. 599-644 | MR | Zbl

[O’G08] K. G. O’Grady - “Irreducible symplectic 4-folds numerically equivalent to (K3) [2] ”, Commun. Contemp. Math. 10 (2008) no. 4, p. 553-608 | DOI | MR | Zbl

[O’G17] K. G. O’Grady - “Covering families of Lagrangian subvarieties” (2017), preprint

[PPZ22] A. Perry, L. Pertusi & X. Zhao - “Stability conditions and moduli spaces for Kuznetsov components of Gushel-Mukai varieties”, Geom. Topol. 26 (2022) no. 7, p. 3055-3121 | Zbl | DOI | MR

[Pro76] C. Procesi - “The invariant theory of n×n matrices”, Adv. Math. 19 (1976) no. 3, p. 306-381 | DOI | MR | Zbl

[PVdB19] A. Polishchuk & M. Van den Bergh - “Semiorthogonal decompositions of the categories of equivariant coherent sheaves for some reflection groups”, J. Eur. Math. Soc. (JEMS) 21 (2019) no. 9, p. 2653-2749 | DOI | MR | Zbl

[Sta25] The Stacks Project Authors - “Stacks Project”, https://stacks.math.columbia.edu, 2025

[Taj23] T. Tajakka - “Uhlenbeck compactification as a Bridgeland moduli space”, Internat. Math. Res. Notices (2023) no. 6, p. 4952-4997 | DOI | MR | Zbl

[Yos01] K. Yoshioka - “Moduli spaces of stable sheaves on abelian surfaces”, Math. Ann. 321 (2001) no. 4, p. 817-884 | DOI | MR | Zbl

Cité par Sources :