×

Hodge numbers of O’Grady 6 via Ngô strings. (English) Zbl 07873607

Summary: We give an alternative computation of the Betti and Hodge numbers for manifolds of \(OG6\) type using the method of Ngô Strings introduced by de Cataldo, Rapagnetta, and Saccà.

MSC:

14J42 Holomorphic symplectic varieties, hyper-Kähler varieties
14F40 de Rham cohomology and algebraic geometry
14F45 Topological properties in algebraic geometry

References:

[1] Beauville, A.: Prym varieties and the Schottky problem. Invent. Math. 41(2), 149-196 (1977) · Zbl 0333.14013
[2] Beauville, A.: Complex algebraic surfaces, volume 34 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge, second edition, Translated from the 1978 French original by R. Barlow, with assistance from N. I. Shepherd-Barron and M. Reid (1996)
[3] Ngo B.C.: Le lemme fondamental pour les algebres de lie 0801, 0446 (2008)
[4] de Cataldo, M.A.: Perverse sheaves and the topology of algebraic varieties. In Geometry of moduli spaces and representation theory, volume 24 of IAS/Park City Math. Ser., pages 1-58. Amer. Math. Soc., Providence, RI, (2017) · Zbl 1439.14060
[5] A support theorem for the Hitchin fibration: the case of \({\rm SL}_n\), Compos. Math., 153, 6, 1316-1347, 2017 · Zbl 1453.14029 · doi:10.1112/S0010437X17007096
[6] de Cataldo, M.; Andrea, A.; Migliorini, L., The decomposition theorem, perverse sheaves and the topology of algebraic maps, Bull. Amer. Math. Soc. N. S., 46, 4, 535-633, 2009 · Zbl 1181.14001 · doi:10.1090/S0273-0979-09-01260-9
[7] Dolgachev, I.: Lectures on invariant theory. London Mathematical Society Lecture Note Series, vol. 296. Cambridge University Press, Cambridge (2003) · Zbl 1023.13006
[8] de Cataldo, Mark A.A., Rapagnetta, A., Saccà, G.: The Hodge numbers of O’Grady 10 via Ngô strings. In J. Math. Pures Appl. (9), 156, 125-178 (2021) · Zbl 1483.14071
[9] Fulton, W., Harris, J.: Representation theory, volume 129 of Graduate Texts in Mathematics. Springer-Verlag, New York, A first course, Readings in Mathematics (1991) · Zbl 0744.22001
[10] Fulton, W.: Intersection theory, volume 2 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer-Verlag, Berlin, second edition, (1998) · Zbl 0885.14002
[11] Green, M., Kim, Y-J., Laza, R, Robles, C.: The llv decomposition of hyper-kaehler cohomology 1906, 03432 (2020)
[12] Göttsche, L.; Soergel, W., Perverse sheaves and the cohomology of Hilbert schemes of smooth algebraic surfaces, Math. Ann., 296, 2, 235-245, 1993 · Zbl 0789.14002 · doi:10.1007/BF01445104
[13] Huybrechts, D., Lehn, M.: The geometry of moduli spaces of sheaves. Aspects of Mathematics, E31. Friedr. Vieweg & Sohn, Braunschweig, (1997)
[14] Kaledin, D., Symplectic singularities from the Poisson point of view, J. Reine Angew. Math., 600, 135-156, 2006 · Zbl 1121.53056
[15] Kollár, J., Mori, S.: Birational geometry of algebraic varieties, volume 134 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, With the collaboration of C. H. Clemens and A. Corti, Translated from the 1998 Japanese original (1998) · Zbl 0926.14003
[16] Kumar, A., Elliptic fibrations on a generic Jacobian Kummer surface, J. Algebraic Geom., 23, 4, 599-667, 2014 · Zbl 1304.14045 · doi:10.1090/S1056-3911-2014-00620-2
[17] Le Potier, J., Faisceaux semi-stables de dimension \(1\) sur le plan projectif, Rev. Roumaine Math. Pures Appl., 38, 7-8, 635-678, 1993 · Zbl 0815.14029
[18] Matsumura, H.: Commutative ring theory, volume 8 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, Translated from the Japanese by M. Reid (1986) · Zbl 0603.13001
[19] Matsushita, D.: Equidimensionality of Lagrangian fibrations on holomorphic symplectic manifolds. Math. Res. Lett. 7(4), 389-391 (2000) · Zbl 1002.53050
[20] Matsushita, D.: Addendum: “On fibre space structures of a projective irreducible symplectic manifold” [Topology 38 (1999), no. 1, 79-83; MR1644091 (99f:14054)]. Topology, 40(2):431-432, (2001) · Zbl 0932.32027
[21] Mehran, A., Kummer surfaces associated to \((1,2)\)-polarized abelian surfaces, Nagoya Math. J., 202, 127-143, 2011 · Zbl 1223.14045 · doi:10.1215/00277630-1260477
[22] Mongardi, G.; Rapagnetta, A.; Saccà, G., The Hodge diamond of O’Grady’s six-dimensional example, Compos. Math., 154, 5, 984-1013, 2018 · Zbl 1420.14095 · doi:10.1112/S0010437X1700803X
[23] Mukai, S., Symplectic structure of the moduli space of sheaves on an abelian or \(K3\) surface, Invent. Math., 77, 1, 101-116, 1984 · Zbl 0565.14002 · doi:10.1007/BF01389137
[24] Mumford, D.: Prym varieties. I. In Contributions to analysis (a collection of papers dedicated to Lipman Bers), pages 325-350. (1974) · Zbl 0299.14018
[25] Le lemme fondamental pour les algèbres de Lie, Publ. Math. Inst. Hautes Études Sci., 111, 1-169, 2010 · Zbl 1200.22011 · doi:10.1007/s10240-010-0026-7
[26] Narasimhan, MS; Ramanan, S., Moduli of vector bundles on a compact Riemann surface, Ann. of Math., 2, 89, 14-51, 1969 · Zbl 0186.54902 · doi:10.2307/1970807
[27] Rapagnetta, A., Topological invariants of O’Grady’s six dimensional irreducible symplectic variety, Math. Z., 256, 1, 1-34, 2007 · Zbl 1121.14014 · doi:10.1007/s00209-006-0022-2
[28] Saito, M., Mixed Hodge modules, Publ. Res. Inst. Math. Sci., 26, 2, 221-333, 1990 · Zbl 0727.14004 · doi:10.2977/prims/1195171082
[29] Schnell, C.: An overview of morihiko saito’s theory of mixed hodge modules 1405, 3096 (2014)
[30] Artin, M., Bertin, J.-E., Demazure, M., Grothendieck, A., Gabriel, P., Raynaud, M., Serre, J.-P.: Schémas en groupes. Séminaire de Géométrie Algébrique de l’Institut des Hautes Études Scientifiques. Institut des Hautes Études Scientifiques, Paris, (1963/1966) · Zbl 0212.52801
[31] Artin, M., Grothendieck, A., Verdier, J.-L.: Theorie de Topos et Cohomologie Etale des Schemas I, II, III, volume 269, 270, 305 of Lecture Notes in Mathematics. Springer, (1971)
[32] Shen, J.; Yin, Q., Topology of lagrangian fibrations and hodge theory of hyper-kähler manifolds, 1812, 10673, 2019
[33] Yoshioka, K., Moduli spaces of stable sheaves on abelian surfaces, Math. Ann., 321, 4, 817-884, 2001 · Zbl 1066.14013 · doi:10.1007/s002080100255
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.