Abstract
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.
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References
Beauville, A.: Prym varieties and the Schottky problem. Invent. Math. 41(2), 149–196 (1977)
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)
Ngo B.C.: Le lemme fondamental pour les algebres de lie 0801, 0446 (2008)
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)
Mark Andrea de Cataldo: A support theorem for the Hitchin fibration: the case of \({\rm SL}_n\). Compos. Math. 153(6), 1316–1347 (2017)
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)
Dolgachev, I.: Lectures on invariant theory. London Mathematical Society Lecture Note Series, vol. 296. Cambridge University Press, Cambridge (2003)
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)
Fulton, W., Harris, J.: Representation theory, volume 129 of Graduate Texts in Mathematics. Springer-Verlag, New York, A first course, Readings in Mathematics (1991)
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)
Green, M., Kim, Y-J., Laza, R, Robles, C.: The llv decomposition of hyper-kaehler cohomology 1906, 03432 (2020)
Göttsche, L., Soergel, W.: Perverse sheaves and the cohomology of Hilbert schemes of smooth algebraic surfaces. Math. Ann. 296(2), 235–245 (1993)
Huybrechts, D., Lehn, M.: The geometry of moduli spaces of sheaves. Aspects of Mathematics, E31. Friedr. Vieweg & Sohn, Braunschweig, (1997)
Kaledin, D.: Symplectic singularities from the Poisson point of view. J. Reine Angew. Math. 600, 135–156 (2006)
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)
Kumar, A.: Elliptic fibrations on a generic Jacobian Kummer surface. J. Algebraic Geom. 23(4), 599–667 (2014)
Le Potier, J.: Faisceaux semi-stables de dimension \(1\) sur le plan projectif. Rev. Roumaine Math. Pures Appl. 38(7–8), 635–678 (1993)
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)
Matsushita, D.: Equidimensionality of Lagrangian fibrations on holomorphic symplectic manifolds. Math. Res. Lett. 7(4), 389–391 (2000)
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)
Mehran, A.: Kummer surfaces associated to \((1,2)\)-polarized abelian surfaces. Nagoya Math. J. 202, 127–143 (2011)
Mongardi, G., Rapagnetta, A., Saccà, G.: The Hodge diamond of O’Grady’s six-dimensional example. Compos. Math. 154(5), 984–1013 (2018)
Mukai, S.: Symplectic structure of the moduli space of sheaves on an abelian or \(K3\) surface. Invent. Math. 77(1), 101–116 (1984)
Mumford, D.: Prym varieties. I. In Contributions to analysis (a collection of papers dedicated to Lipman Bers), pages 325–350. (1974)
Bao Châu Ngô: Le lemme fondamental pour les algèbres de Lie. Publ. Math. Inst. Hautes Études Sci. 111, 1–169 (2010)
Narasimhan, M.S., Ramanan, S.: Moduli of vector bundles on a compact Riemann surface. Ann. of Math. 2(89), 14–51 (1969)
Rapagnetta, A.: Topological invariants of O’Grady’s six dimensional irreducible symplectic variety. Math. Z. 256(1), 1–34 (2007)
Saito, M.: Mixed Hodge modules. Publ. Res. Inst. Math. Sci. 26(2), 221–333 (1990)
Schnell, C.: An overview of morihiko saito’s theory of mixed hodge modules 1405, 3096 (2014)
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)
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)
Shen, J., Yin, Q.: Topology of lagrangian fibrations and hodge theory of hyper-kähler manifolds 1812, 10673 (2019)
Yoshioka, K.: Moduli spaces of stable sheaves on abelian surfaces. Math. Ann. 321(4), 817–884 (2001)
Acknowledgements
I would first like to greatly thank my advisor Mark de Cataldo for the many enlightening discussions and his continued support throughout the writing of this paper. I would also like to thank Mads Bach Villadsen for the many useful discussions and for reading several preliminary drafts, Lisa Marquand for the many useful discussions and her support, and Yoonjoo Kim for a useful discussion on the LLV decomposition. I would like to give a special thanks to Antonio Rapagnetta for an extremely useful discussion on monodromy (in particular, see Lemma 3.1.3 and Section 3.2). Finally, I thank the reviewer for their comments and suggestions improving this manuscript.
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