The global moduli theory of symplectic varieties

References

[1] E. Amerik and M. Verbitsky, Contraction centers in families of hyperkähler manifolds, Selecta Math. (N. S.) 27 (2021), no. 4, Paper No. 60. 10.1007/s00029-021-00677-8Suche in Google Scholar

[2] V. Ancona and B. Gaveau, Differential forms on singular varieties. De Rham and Hodge theory simplified, Pure Appl. Math. (Boca Raton) 273, Chapman & Hall/CRC, Boca Raton 2006. 10.1201/9781420026528Suche in Google Scholar

[3] F. Anella and A. Höring, Twisted cotangent bundle of hyperkähler manifolds (with an appendix by Simone Diverio), J. Éc. polytech. Math. 8 (2021), 1429–1457. 10.5802/jep.175Suche in Google Scholar

[4] M. Artin, On the solutions of analytic equations, Invent. Math. 5 (1968), 277–291. 10.1007/BF01389777Suche in Google Scholar

[5] B. Bakker and C. Lehn, A global Torelli theorem for singular symplectic varieties, J. Eur. Math. Soc. (JEMS) 23 (2021), no. 3, 949–994. 10.4171/JEMS/1026Suche in Google Scholar

[6] A. Beauville, Variétés Kähleriennes dont la première classe de Chern est nulle, J. Differential Geom. 18 (1983), no. 4, 755–782. 10.4310/jdg/1214438181Suche in Google Scholar

[7] A. Beauville, Symplectic singularities, Invent. Math. 139 (2000), no. 3, 541–549. 10.1007/s002229900043Suche in Google Scholar

[8] E. Bierstone and P. D. Milman, Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Invent. Math. 128 (1997), no. 2, 207–302. 10.1007/s002220050141Suche in Google Scholar

[9] 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 (2010), no. 2, 405–468. 10.1090/S0894-0347-09-00649-3Suche in Google Scholar

[10] F. A. Bogomolov, On the cohomology ring of a simple hyper-Kähler manifold (on the results of Verbitsky), Geom. Funct. Anal. 6 (1996), no. 4, 612–618. 10.1007/BF02247113Suche in Google Scholar

[11] A. Borel, Some metric properties of arithmetic quotients of symmetric spaces and an extension theorem, J. Differential Geom. 6 (1972), 543–560. 10.4310/jdg/1214430642Suche in Google Scholar

[12] S. Boucksom, Divisorial Zariski decompositions on compact complex manifolds, Ann. Sci. Éc. Norm. Supér. (4) 37 (2004), no. 1, 45–76. 10.1016/j.ansens.2003.04.002Suche in Google Scholar

[13] C. Camere, Symplectic involutions of holomorphic symplectic four-folds, Bull. Lond. Math. Soc. 44 (2012), no. 4, 687–702. 10.1112/blms/bdr133Suche in Google Scholar

[14] C. Camere, A. Garbagnati, G. Kapustka and M. Kapustka, Projective models of Nikulin orbifolds, preprint (2021), http://arxiv.org/abs/2104.09234. Suche in Google Scholar

[15] T. C. Collins and V. Tosatti, A singular Demailly–Păun theorem, C. R. Math. Acad. Sci. Paris 354 (2016), no. 1, 91–95. 10.1016/j.crma.2015.10.012Suche in Google Scholar

[16] O. Debarre, Higher-dimensional algebraic geometry, Universitext, Springer, New York 2001. 10.1007/978-1-4757-5406-3Suche in Google Scholar

[17] P. Deligne, Théorème de Lefschetz et critères de dégénérescence de suites spectrales, Publ. Math. Inst. Hautes Études Sci. 35 (1968), 259–278. 10.1007/BF02698925Suche in Google Scholar

[18] P. Deligne, P. Griffiths, J. Morgan and D. Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975), no. 3, 245–274. 10.1007/BF01389853Suche in Google Scholar

[19] J.-P. Demailly, Regularization of closed positive currents and intersection theory, J. Algebraic Geom. 1 (1992), no. 3, 361–409. Suche in Google Scholar

[20] J.-P. Demailly, Analytic methods in algebraic geometry, Surv. Mod. Math. 1, International Press, Somerville 2012. Suche in Google Scholar

[21] J.-P. Demailly and M. Paun, Numerical characterization of the Kähler cone of a compact Kähler manifold, Ann. of Math. (2) 159 (2004), no. 3, 1247–1274. 10.4007/annals.2004.159.1247Suche in Google Scholar

[22] A. Douady, Le problème des modules locaux pour les espaces 𝐂 -analytiques compacts, Ann. Sci. Éc. Norm. Supér. (4) 7 (1974), 569–602. 10.24033/asens.1280Suche in Google Scholar

[23] S. Druel, Quelques remarques sur la décomposition de Zariski divisorielle sur les variétés dont la première classe de Chern est nulle, Math. Z. 267 (2011), no. 1–2, 413–423. 10.1007/s00209-009-0626-4Suche in Google Scholar

[24] S. Druel, A decomposition theorem for singular spaces with trivial canonical class of dimension at most five, Invent. Math. 211 (2018), no. 1, 245–296. 10.1007/s00222-017-0748-ySuche in Google Scholar

[25] S. Druel and H. Guenancia, A decomposition theorem for smoothable varieties with trivial canonical class, J. Éc. polytech. Math. 5 (2018), 117–147. 10.5802/jep.65Suche in Google Scholar

[26] H. Flenner and S. Kosarew, On locally trivial deformations, Publ. Res. Inst. Math. Sci. 23 (1987), no. 4, 627–665. 10.2977/prims/1195176251Suche in Google Scholar

[27] L. Fu and G. Menet, On the Betti numbers of compact holomorphic symplectic orbifolds of dimension four, Math. Z. 299 (2021), no. 1–2, 203–231. 10.1007/s00209-020-02682-7Suche in Google Scholar

[28] A. Fujiki, On automorphism groups of compact Kähler manifolds, Invent. Math. 44 (1978), no. 3, 225–258. 10.1007/BF01403162Suche in Google Scholar

[29] A. Fujiki, On primitively symplectic compact Kähler V-manifolds of dimension four, Classification of algebraic and analytic manifolds (Katata 1982), Progr. Math. 39, Birkhäuser, Boston (1983), 71–250. Suche in Google Scholar

[30] A. Fujiki, On the de Rham cohomology group of a compact Kähler symplectic manifold, Algebraic geometry (Sendai 1985), Adv. Stud. Pure Math. 10, North-Holland, Amsterdam (1987), 105–165. 10.2969/aspm/01010105Suche in Google Scholar

[31] W. Fulton, Intersection theory, 2nd ed., Ergeb. Math. Grenzgeb. (3) 2, Springer, Berlin 1998. 10.1007/978-1-4612-1700-8Suche in Google Scholar

[32] H. Grauert, Über Modifikationen und exzeptionelle analytische Mengen, Math. Ann. 146 (1962), 331–368. 10.1007/BF01441136Suche in Google Scholar

[33] H. Grauert, Der Satz von Kuranishi für kompakte komplexe Räume, Invent. Math. 25 (1974), 107–142. 10.1007/BF01390171Suche in Google Scholar

[34] D. Greb, H. Guenancia and S. Kebekus, Klt varieties with trivial canonical class: Holonomy, differential forms, and fundamental groups, Geom. Topol. 23 (2019), no. 4, 2051–2124. 10.2140/gt.2019.23.2051Suche in Google Scholar

[35] D. Greb, S. Kebekus, S. J. Kovács and T. Peternell, Differential forms on log canonical spaces, Publ. Math. Inst. Hautes Études Sci. 114 (2011), 87–169. 10.1007/s10240-011-0036-0Suche in Google Scholar

[36] D. Greb, S. Kebekus and T. Peternell, Singular spaces with trivial canonical class, Minimal models and extremal rays (Kyoto 2011), Adv. Stud. Pure Math. 70, Mathematical Society of Japan, Tokyo (2016), 67–113. 10.2969/aspm/07010067Suche in Google Scholar

[37] P. Griffiths and J. Harris, Principles of algebraic geometry, Wiley Classics Lib., John Wiley & Sons, New York 1994. 10.1002/9781118032527Suche in Google Scholar

[38] M. Gross, D. Huybrechts and D. Joyce, Calabi–Yau manifolds and related geometries, Universitext, Springer, Berlin 2003. 10.1007/978-3-642-19004-9Suche in Google Scholar

[39] H. Guenancia, Semistability of the tangent sheaf of singular varieties, Algebr. Geom. 3 (2016), no. 5, 508–542. 10.14231/AG-2016-024Suche in Google Scholar

[40] C. D. Hacon and J. Mckernan, On Shokurov’s rational connectedness conjecture, Duke Math. J. 138 (2007), no. 1, 119–136. 10.1215/S0012-7094-07-13813-4Suche in Google Scholar

[41] R. Hartshorne, Generalized divisors on Gorenstein schemes, K-Theory 8 (1994), no. 3, 287–339. 10.1007/BF00960866Suche in Google Scholar

[42] A. Höring and T. Peternell, Algebraic integrability of foliations with numerically trivial canonical bundle, Invent. Math. 216 (2019), no. 2, 395–419. 10.1007/s00222-018-00853-2Suche in Google Scholar

[43] D. Huybrechts, Birational symplectic manifolds and their deformations, J. Differential Geom. 45 (1997), no. 3, 488–513. 10.4310/jdg/1214459840Suche in Google Scholar

[44] D. Huybrechts, Compact hyper-Kähler manifolds: Basic results, Invent. Math. 135 (1999), no. 1, 63–113. 10.1007/s002220050280Suche in Google Scholar

[45] D. Huybrechts, Erratum: “Compact hyper-Kähler manifolds: Basic results” [Invent. Math. 135 (1999), no. 1, 63–113; MR1664696 (2000a:32039)], Invent. Math. 152 (2003), no. 1, 209–2012. 10.1007/s00222-002-0280-5Suche in Google Scholar

[46] D. Huybrechts, The Kähler cone of a compact hyperkähler manifold, Math. Ann. 326 (2003), no. 3, 499–513. 10.1007/s00208-003-0433-xSuche in Google Scholar

[47] D. Huybrechts, A global Torelli theorem for hyperkähler manifolds [after M. Verbitsky], Séminaire Bourbaki. Vol. 2010/2011. Exposés 1027–1042, Astérisque 348, Société Mathématique de France, Paris (2012), 375–403. Suche in Google Scholar

[48] D. Huybrechts, Finiteness of polarized K3 surfaces and hyperkähler manifolds, Ann. H. Lebesgue 1 (2018), 227–246. 10.5802/ahl.6Suche in Google Scholar

[49] S. Ji and B. Shiffman, Properties of compact complex manifolds carrying closed positive currents, J. Geom. Anal. 3 (1993), no. 1, 37–61. 10.1007/BF02921329Suche in Google Scholar

[50] D. Kaledin, Symplectic singularities from the Poisson point of view, J. reine angew. Math. 600 (2006), 135–156. 10.1515/CRELLE.2006.089Suche in Google Scholar

[51] D. Kaledin, M. Lehn and C. Sorger, Singular symplectic moduli spaces, Invent. Math. 164 (2006), no. 3, 591–614. 10.1007/s00222-005-0484-6Suche in Google Scholar

[52] Y. Kawamata, Unobstructed deformations. A remark on a paper of Z. Ran: “Deformations of manifolds with torsion or negative canonical bundle” [J. Algebraic Geom. 1 (1992), no. 2, 279–291; MR1144440 (93e:14015)], J. Algebraic Geom. 1 (1992), no. 2, 183–190. Suche in Google Scholar

[53] Y. Kawamata, Erratum on: Unobstructed deformations. A remark on a paper of Z. Ran: “Deformations of manifolds with torsion or negative canonical bundle” [J. Algebraic Geom. 1 (1992), no. 2, 183–190; MR1144434 (93e:14016)], J. Algebraic Geom. 6 (1997), no. 4, 803–804. Suche in Google Scholar

[54] S. Kebekus and C. Schnell, Extending holomorphic forms from the regular locus of a complex space to a resolution of singularities, J. Amer. Math. Soc. 34 (2021), no. 2, 315–368. 10.1090/jams/962Suche in Google Scholar

[55] T. Kirschner, Period mappings with applications to symplectic complex spaces, Lecture Notes in Math. 2140, Springer, Cham 2015. 10.1007/978-3-319-17521-8Suche in Google Scholar

[56] J. Kollár, Projectivity of complete moduli, J. Differential Geom. 32 (1990), no. 1, 235–268. 10.4310/jdg/1214445046Suche in Google Scholar

[57] J. Kollár, R. Laza, G. Saccà and C. Voisin, Remarks on degenerations of hyper-Kähler manifolds, Ann. Inst. Fourier (Grenoble) 68 (2018), no. 7, 2837–2882. 10.5802/aif.3228Suche in Google Scholar

[58] J. Kollár and S. Mori, Classification of three-dimensional flips, J. Amer. Math. Soc. 5 (1992), no. 3, 533–703. 10.1090/S0894-0347-1992-1149195-9Suche in Google Scholar

[59] J. Kollár and S. Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Math. 134, Cambridge University, Cambridge 1998. 10.1017/CBO9780511662560Suche in Google Scholar

[60] C. Lehn, Symplectic lagrangian fibrations, Dissertation, Johannes–Gutenberg–Universität Mainz, 2011, http://doi.org/10.25358/openscience-3153. Suche in Google Scholar

[61] C. Lehn, Deformations of Lagrangian subvarieties of holomorphic symplectic manifolds, Math. Res. Lett. 23 (2016), no. 2, 473–497. 10.4310/MRL.2016.v23.n2.a9Suche in Google Scholar

[62] C. Lehn, Twisted cubics on singular cubic fourfolds – on Starr’s fibration, Math. Z. 290 (2018), no. 1–2, 379–388. 10.1007/s00209-017-2021-xSuche in Google Scholar

[63] C. Lehn and G. Pacienza, Deformations of singular symplectic varieties and termination of the log minimal model program, Algebr. Geom. 3 (2016), no. 4, 392–406. 10.14231/AG-2016-018Suche in Google Scholar

[64] T. Matsusaka, On polarized normal varieties. I, Nagoya Math. J. 104 (1986), 175–211. 10.1017/S0027763000022741Suche in Google Scholar

[65] D. Matsushita, Fujiki relation on symplectic varieties, preprint (2001), https://arxiv.org/abs/math/0109165. Suche in Google Scholar

[66] D. Matsushita, On base manifolds of Lagrangian fibrations, Sci. China Math. 58 (2015), no. 3, 531–542. 10.1007/s11425-014-4927-7Suche in Google Scholar

[67] G. Menet, Global Torelli theorem for irreducible symplectic orbifolds, J. Math. Pures Appl. (9) 137 (2020), 213–237. 10.1016/j.matpur.2020.03.010Suche in Google Scholar

[68] G. Mongardi, On symplectic automorphisms of hyper-Kähler fourfolds of K3 [ 2 ] type, Michigan Math. J. 62 (2013), no. 3, 537–550. 10.1307/mmj/1378757887Suche in Google Scholar

[69] Y. Namikawa, A note on symplectic singularities, 2001. 10.1515/crll.2001.070Suche in Google Scholar

[70] Y. Namikawa, Deformation theory of singular symplectic n-folds, Math. Ann. 319 (2001), no. 3, 597–623. 10.1007/PL00004451Suche in Google Scholar

[71] Y. Namikawa, Extension of 2-forms and symplectic varieties, J. reine angew. Math. 539 (2001), 123–147. 10.1515/crll.2001.070Suche in Google Scholar

[72] Y. Namikawa, Projectivity criterion of Moishezon spaces and density of projective symplectic varieties, Internat. J. Math. 13 (2002), no. 2, 125–135. 10.1142/S0129167X02001277Suche in Google Scholar

[73] Y. Namikawa, On deformations of ℚ -factorial symplectic varieties, J. reine angew. Math. 599 (2006), 97–110. 10.1515/CRELLE.2006.079Suche in Google Scholar

[74] V. V. Nikulin, Finite groups of automorphisms of Kählerian surfaces of type K ⁢ 3 , Uspehi Mat. Nauk 31 (1976), no. 2(188), 223–224. Suche in Google Scholar

[75] Y. Odaka and Y. Oshima, Collapsing K3 surfaces, tropical geometry and moduli compactifications of Satake, Morgan–Shalen type, MSJ Memoirs 40, Mathematical Society of Japan, Tokyo 2021. 10.1142/e071Suche in Google Scholar

[76] K. G. O’Grady, Desingularized moduli spaces of sheaves on a K ⁢ 3 , J. reine angew. Math. 512 (1999), 49–117. 10.1515/crll.1999.056Suche in Google Scholar

[77] M. Paun, Sur l’effectivité numérique des images inverses de fibrés en droites, Math. Ann. 310 (1998), no. 3, 411–421. 10.1007/s002080050154Suche in Google Scholar

[78] C. A. M. Peters and J. H. M. Steenbrink, Mixed Hodge structures, Ergeb. Math. Grenzgeb. (3) 52, Springer, Berlin 2008. Suche in Google Scholar

[79] Z. Ran, Deformations of manifolds with torsion or negative canonical bundle, J. Algebraic Geom. 1 (1992), no. 2, 279–291. Suche in Google Scholar

[80] G. Saccà, Compactification of Lagrangian fibrations, to appear. Suche in Google Scholar

[81] M. Saito, Modules de Hodge polarisables, Publ. Res. Inst. Math. Sci. 24 (1988), no. 6, 849–995. 10.2977/prims/1195173930Suche in Google Scholar

[82] M. Saito, Decomposition theorem for proper Kähler morphisms, Tohoku Math. J. (2) 42 (1990), no. 2, 127–147. 10.2748/tmj/1178227650Suche in Google Scholar

[83] J. Sawon, Isotrivial elliptic K3 surfaces and Lagrangian fibrations, preprint (2014), http://arxiv.org/abs/1406.1233. Suche in Google Scholar

[84] M. Schwald, Fujiki relations and fibrations of irreducible symplectic varieties, Épijournal Géom. Algébrique 4 (2020), Article ID 7. 10.46298/epiga.2020.volume4.4557Suche in Google Scholar

[85] E. Sernesi, Deformations of algebraic schemes, Grundlehren Math. Wiss. 334, Springer, Berlin 2006. Suche in Google Scholar

[86] J. Varouchas, Kähler spaces and proper open morphisms, Math. Ann. 283 (1989), no. 1, 13–52. 10.1007/BF01457500Suche in Google Scholar

[87] M. Verbitsky, Cohomology of compact hyper-Kähler manifolds and its applications, Geom. Funct. Anal. 6 (1996), no. 4, 601–611. 10.1007/BF02247112Suche in Google Scholar

[88] M. Verbitsky, Mapping class group and a global Torelli theorem for hyperkähler manifolds, Duke Math. J. 162 (2013), no. 15, 2929–2986. 10.1215/00127094-2382680Suche in Google Scholar

[89] M. Verbitsky, Ergodic complex structures on hyperkähler manifolds, Acta Math. 215 (2015), no. 1, 161–182. 10.1007/s11511-015-0131-zSuche in Google Scholar

[90] M. Verbitsky, Ergodic complex structures on hyperkahler manifolds: An erratum, preprint (2017), http://arxiv.org/abs/1708.05802. 10.1007/s11511-015-0131-zSuche in Google Scholar

[91] M. Verbitsky, Mapping class group and global Torelli theorem for hyperkahler manifolds: An erratum, preprint (2019), http://arxiv.org/abs/1908.11772. 10.1215/00127094-2382680Suche in Google Scholar

[92] O. Villamayor, Constructiveness of Hironaka’s resolution, Ann. Sci. Éc. Norm. Supér. (4) 22 (1989), no. 1, 1–32. 10.24033/asens.1573Suche in Google Scholar

[93] Martin M. W, MathOverflow: Is ℝ 3 ∖ ℚ 3 simply connected? 2015, https://mathoverflow.net/questions/215923/is-mathbbr3-setminus-mathbbq3-simply-connected. Suche in Google Scholar