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Published in:
Negative Rational Curves and Their Deformations on Hyperkähler Manifolds Read chapter Read first chapter

2020 | OriginalPaper | Chapter

Examples of Irreducible Symplectic Varieties

Author : Arvid Perego

Published in: Birational Geometry and Moduli Spaces

Publisher: Springer International Publishing

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Abstract

Irreducible symplectic manifolds are one of the three building blocks of compact Kähler manifolds with numerically trivial canonical bundle by the Beauville-Bogomolov decomposition theorem. There are several singular analogues of irreducible symplectic manifolds, in particular in the context of compact Kähler orbifolds, and in the context of normal projective varieties with canonical singularities. In this paper we will collect their definitions, analyze their mutual relations and provide a list of known examples.

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Literature
1.
Bakker, B., Lehn, C.: A Global Torelli Theorem for Singular Symplectic Varieties (2016). Preprint arxiv:1612.07894
2.
Beauville, A.: Variétés kähleriennes dont la première classe de Chern est nulle. J. Differ. Geom. 18, 755–782 (1983)MATHCrossRef
3.
4.
Beauville, A., Donagi, R.: La variété des droites d’une hypersurface cubique de dimension 4. C. R. Acad. Sci. Paris Sr. I Math. 301(14), 703–706 (1985)MATH
5.
Blache, R.: Chern classes and Hirzeburch-Riemann-Roch theorem for coherent sheaves on complex projective orbifolds with isolated singularities. Math. Z. 222, 7–57 (1996)MathSciNetMATHCrossRef
6.
Bogomolov, F.: On the decomposition of Kähler manifolds with trivial canonical class. Math. USSR-Sb. 22, 580–583 (1974)MATHCrossRef
7.
8.
Campana, F.: Orbifoldes à première classe de Chern nulle. In: The Fano Conference, pp. 339–351, University Torino, Torino (2004)
9.
Debarre, O., Voisin, C.: Hyper-Kähler fourfolds and Grassmann Geometry. J. Reine Angew. Math. 649, 63–87 (2010)MathSciNetMATH
10.
Demailly, J.-P., Kollár, J.: Semi-continuity of Complex Singularity Exponents and Kähler-Einstein Metrics on Fano Orbifolds. Ann. Sci. Ec. Norm. Sup. Série 4 34(4), 525–556 (2001)MathSciNetMATHCrossRef
11.
Druel, S.: A decomposition theorem for singular spaces with trivial canonical class of dimension at most five. Invent. Math. 211(1), 245–296 (2018)MathSciNetMATHCrossRef
12.
Druel, S., Guenancia, H.: A decomposition theorem for smoothable varieties with trivial canonical class. J. de l’École Polytechnique—Mathématiques 5, 117–147 (2018)MathSciNetMATHCrossRef
13.
Eyssidieux, P., Guedj, V., Zeriahi, A.: Singular Kähler-Einstein metrics. J. Am. Math. Soc. 22(3), 607–639 (2009)MATHCrossRef
14.
15.
Fujiki, A.: On primitively symplectic compact Kähler V-Manifolds of dimension four. In: Classification of Algebraic and Analytic Manifolds. Progress in Mathematics 39, 71–250 (1983)MATH
16.
Greb, D., Guenancia, H., Kebekus, S.: Klt with Trivial Canonical Class—Holonomy, Differential Forms, and Fundamental Group. Geom. Topol. 23, 2051–2124 (2019)MathSciNetMATHCrossRef
17.
Greb, D., Kebekus, S., Kovács, S., Peternell, T.: Differential forms on log-canonical spaces. Publ. Math. IHES 114(1), 87–169 (2011)MathSciNetMATHCrossRef
18.
Greb, D., Kebekus, S., Peternell, T.: Singular spaces with trivial canonical class. In: Proceedings of the conference in honor of Shigefumi Mori’s 60th birthday, minimal models and extreme rays. Advanced Studies in Pure Mathematics. Kinokuniya Publishing House, Tokyo (2011)
19.
Höring, A., Peternell, T.: Algebraic Integrability of Foliations with Numerically Trivial Canonical Bundle. Invent. Math. 216, 395–419 (2019)MathSciNetMATHCrossRef
20.
Huybrechts, D., Lehn, M.: The Geometry of Moduli Spaces of Sheaves. Aspects of Mathematics E, vol. 31. Vieweg, Braunschweig (1997)MATHCrossRef
21.
Huybrechts, D., Nieper-Wisskirchen, M.: Remarks on derived equivalence of Ricci-flat manifolds. Math. Z. 267(3–4), 939–963 (2011)MathSciNetMATHCrossRef
22.
Iliev, A., Ranestad, K.: K3 surfaces of genus 8 and varieties of sums of powers of cubic fourfolds. Trans. Am. Math. Soc. 353, 1455–1468 (2001)MathSciNetMATHCrossRef
23.
24.
Kamenova, L., Mongardi, G., Oblomkov, A.: Symplectic involutions of K3[n] type and Kummer n type manifolds (2018). Preprint arxiv:1809.02810
25.
Kapfer, S., Menet, G.: Integral cohomology of the generalized Kummer fourfold. Algebr. Geom. 5(5), 523–567 (2018)MathSciNetMATH
26.
Kawamata, Y.: Minimal models and the Kodaira dimension of algebraic fiber spaces. J. Reine Angew. Math. 363, 1–46 (1985)MathSciNetMATH
27.
Kollár, J., Mori, S.: Birational geometry of algebraic varieties. In: Cambridge Tracts in Mathematics, vol. 134. Cambridge University, Cambridge (1998)
28.
Lehn, C., Lehn, M., Sorger, C., van Straten, D.: Twisted cubics on cubic fourfolds. J. Reine Angew. Math. 731, 87–128 (2017)MathSciNetMATH
29.
Lehn, M., Sorger, C.: La singularité de O’Grady. J. Algebraic Geom. 15, 756–770 (2006)MATHCrossRef
30.
Markushevich, D., Tikhomirov, A.S.: New symplectic V-manifolds of dimension four via the relative compactified Prymian. Int. J. Math. 18(10), 1187–1224 (2007)MathSciNetMATHCrossRef
31.
Matteini, T.: A new singular symplectic variety of dimension 6 with a Lagrangian Prym fibration. Manuscripta Math. 149(1), 131–151 (2016)MathSciNetMATHCrossRef
32.
Menet, G.: Beauville-Bogomolov lattice for a singular symplectic variety of dimension 4. J. Pure App. Math. 219(5), 1455–1495 (2014)MathSciNetMATH
33.
34.
Menet, G.: A Global Torelli Theorem for irreducible symplectic orbifolds. Preprint arxiv:1805.12426
35.
Menet, G.: Integral cohomology of quotients via toric geometry (2019). Preprint
36.
Mongardi, G.: Automorphisms of Hyperkähler Manifolds. PhD Thesis arxiv:1303.4670
37.
38.
Mukai, S.: On the moduli space of bundles on K3 surfaces I. In: Vector Bundles on Algebraic Varieties, Bombay (1984). Tata Institute of Fundamental Research Studies on Mathematics, vol. 11, , pp. 341–413. Tata Institute of Fundamental Research, Bombay (1987)
39.
Namikawa, Y.: A Note on Symplectic Singularities (2001). Preprint arxiv:0101028
40.
Namikawa, Y.: Extension of 2-forms and symplectic varieties. J. Reine Angew. Math. 539, 123–147 (2001)MathSciNetMATH
41.
Namikawa, Y.: On deformations of \(\mathbb {Q}\)-factorial symplectic varieties. J. Reine Angew. Math. 599, 97–110 (2006)
42.
O’Grady, K.: The Weight-two Hodge Structure of Moduli Spaces of Sheaves on a K3 Surface. J. Algebraic Geom. 6(4), 599–644 (1997)MathSciNetMATH
43.
44.
45.
46.
47.
Perego, A., Rapagnetta, A.: The moduli spaces of sheaves on K3 surfaces are irreducible symplectic varieties. Preprint arxiv:1802.01182
48.
49.
Schwald, M.: Fujiki Relations and Fibrations of Irreducible Symplectic Varities (2017). Preprint arxiv:1701.09069
50.
51.
Yau, S.-T.: On the ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. Commun. Pure App. Math 31, 339–411 (1978)MATHCrossRef
52.
Yoshioka, K.: Irreducibility of Moduli Spaces of Vector Bundles on K3 Surfaces (1999). Preprint arxiv:math/9907001
53.
54.
Yoshioka, K.: Twisted stability and Fourier-Mukai functor I. Compos. Math. 138(3), 261–288 (2003)MATHCrossRef
55.
Metadata
Title
Examples of Irreducible Symplectic Varieties
Author
Arvid Perego
Copyright Year
2020
Book
Birational Geometry and Moduli Spaces

Print ISBN: 978-3-030-37113-5

Electronic ISBN: 978-3-030-37114-2

Copyright Year: 2020

DOI
https://doi.org/10.1007/978-3-030-37114-2_9

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