Hyperkähler metrics near Lagrangian submanifolds and symplectic groupoids
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Maxence Mayrand
Abstract
The first part of this paper is a generalization of the Feix–Kaledin theorem on the existence of a hyperkähler metric on a neighborhood of the zero section of the cotangent bundle of a Kähler manifold. We show that the problem of constructing a hyperkähler structure on a neighborhood of a complex Lagrangian submanifold in a holomorphic symplectic manifold reduces to the existence of certain deformations of holomorphic symplectic structures. The Feix–Kaledin structure is recovered from the twisted cotangent bundle. We then show that every holomorphic symplectic groupoid over a compact holomorphic Poisson surface of Kähler type has a hyperkähler structure on a neighborhood of its identity section. More generally, we reduce the existence of a hyperkähler structure on a symplectic realization of a holomorphic Poisson manifold of any dimension to the existence of certain deformations of holomorphic Poisson structures adapted from Hitchin’s unobstructedness theorem.
Funding statement: This work was supported by an NSERC Postdoctoral Fellowship and additional support from the FRQNT.
Acknowledgements
I thank Marco Gualtieri for helpful discussions, and the anonymous referee for useful comments.
References
[1] A. Abasheva, Feix–Kaledin metric on the total spaces of cotangent bundles to Kähler quotients, preprint (2020), https://arxiv.org/abs/2007.05773. 10.1093/imrn/rnab047Search in Google Scholar
[2] M. Atiyah and N. Hitchin, The geometry and dynamics of magnetic monopoles, M. B. Porter Lectures, Princeton University Press, Princeton 1988. 10.1515/9781400859306Search in Google Scholar
[3] M. Bailey and M. Gualtieri, Integration of generalized complex structures, preprint (2016), https://arxiv.org/abs/1611.03850. Search in Google Scholar
[4] A. Beauville, Variétés Kähleriennes dont la première classe de Chern est nulle, J. Differential Geom. 18 (1983), no. 4, 755–782. 10.4310/jdg/1214438181Search in Google Scholar
[5] R. Bielawski, Hyper-Kähler structures and group actions, J. Lond. Math. Soc. (2) 55 (1997), no. 2, 400–414. 10.1112/S0024610796004723Search in Google Scholar
[6] O. Biquard, Sur les équations de Nahm et la structure de Poisson des algèbres de Lie semi-simples complexes, Math. Ann. 304 (1996), no. 2, 253–276. 10.1007/BF01446293Search in Google Scholar
[7] O. Biquard and P. Gauduchon, Hyper-Kähler metrics on cotangent bundles of Hermitian symmetric spaces, Geometry and physics (Aarhus 1995), Lecture Notes in Pure and Appl. Math. 184, Dekker, New York (1997), 287–298. 10.1201/9781003072393-23Search in Google Scholar
[8] O. Biquard and P. Gauduchon, Géométrie hyperkählérienne des espaces hermitiens symétriques complexifiés, Séminaire de Théorie Spectrale et Géométrie, Vol. 16, Année 1997–1998, Sémin. Théor. Spectr. Géom. 16, Univ. Grenoble I, Saint-Martin-d’Hères (1998), 127–173. 10.5802/tsg.199Search in Google Scholar
[9] F. Bischoff, M. Gualtieri and M. Zabzine, Morita equivalence and the generalized Kähler potential, preprint (2018), https://arxiv.org/abs/1804.05412. 10.4310/jdg/1659987891Search in Google Scholar
[10] F. Bogomolov, R. Deev and M. Verbitsky, Sections of Lagrangian fibrations on holomorphically symplectic manifolds and degenerate twistorial deformations, preprint (2020), https://arxiv.org/abs/2011.00469. 10.1016/j.aim.2022.108479Search in Google Scholar
[11] D. Broka and P. Xu, Symplectic realizations of holomorphic poisson manifolds, preprint (2015), https://arxiv.org/abs/1512.08847. Search in Google Scholar
[12] E. Calabi, Métriques kählériennes et fibrés holomorphes, Ann. Sci. Éc. Norm. Supér. (4) 12 (1979), no. 2, 269–294. 10.24033/asens.1367Search in Google Scholar
[13] A. S. Cattaneo and G. Felder, Poisson sigma models and symplectic groupoids, Quantization of singular symplectic quotients, Progr. Math. 198, Birkhäuser, Basel (2001), 61–93. 10.1007/978-3-0348-8364-1_4Search in Google Scholar
[14] K. Cieliebak and Y. Eliashberg, From Stein to Weinstein and back. Symplectic geometry of affine complex manifolds, Amer. Math. Soc. Colloq. Publ. 59, American Mathematical Society, Providence 2012. Search in Google Scholar
[15] A. Coste, P. Dazord and A. Weinstein, Groupoïdes symplectiques, Publications du Département de Mathématiques. Nouvelle Série. A, Vol. 2, Publ. Dép. Math. Nouvelle Sér. A 87, Univ. Claude-Bernard, Lyon (1987), i–ii, 1–62. Search in Google Scholar
[16] M. Crainic and R. L. Fernandes, Integrability of Lie brackets, Ann. of Math. (2) 157 (2003), no. 2, 575–620. 10.4007/annals.2003.157.575Search in Google Scholar
[17] M. Crainic and R. L. Fernandes, Integrability of Poisson brackets, J. Differential Geom. 66 (2004), no. 1, 71–137. 10.4310/jdg/1090415030Search in Google Scholar
[18] S. K. Donaldson, Holomorphic discs and the complex Monge–Ampère equation, J. Symplectic Geom. 1 (2002), no. 2, 171–196. 10.4310/JSG.2001.v1.n2.a1Search in Google Scholar
[19] B. Feix, Hyperkähler metrics on cotangent bundles, J. reine angew. Math. 532 (2001), 33–46. 10.1515/crll.2001.017Search in Google Scholar
[20]
B. Feix,
Twistor spaces of hyperkähler manifolds with
[21] D. Greb and M. L. Wong, Canonical complex extensions of Kähler manifolds, J. Lond. Math. Soc. (2) 101 (2020), no. 2, 786–827. 10.1112/jlms.12287Search in Google Scholar
[22] M. Gualtieri, Generalized Kähler metrics from Hamiltonian deformations, Geometry and physics. Vol. II, Oxford University Press, Oxford (2018), 551–579. 10.1093/oso/9780198802020.003.0023Search in Google Scholar
[23] N. Hitchin, Deformations of holomorphic Poisson manifolds, Mosc. Math. J. 12 (2012), no. 3, 567–591, 669. 10.17323/1609-4514-2012-12-3-567-591Search in Google Scholar
[24] N. J. Hitchin, Polygons and gravitons, Math. Proc. Cambridge Philos. Soc. 85 (1979), no. 3, 465–476. 10.1142/9789814539395_0033Search in Google Scholar
[25] N. J. Hitchin, The self-duality equations on a Riemann surface, Proc. Lond. Math. Soc. (3) 55 (1987), no. 1, 59–126. 10.1112/plms/s3-55.1.59Search in Google Scholar
[26] N. J. Hitchin, A. Karlhede, U. Lindström and M. Roček, Hyper-Kähler metrics and supersymmetry, Comm. Math. Phys. 108 (1987), no. 4, 535–589. 10.1007/BF01214418Search in Google Scholar
[27] D. Kaledin, A canonical hyperkähler metric on the total space of a cotangent bundle, Quaternionic structures in mathematics and physics (Rome 1999), Univ. Studi Roma “La Sapienza”, Rome (1999), 195–230. 10.1142/9789812810038_0010Search in Google Scholar
[28] D. Kaledin, Hyperkähler structures on total spaces of holomorphic cotangent bundles, Hyperkähler manifolds, Math. Phys. (Somerville) 12, International Press, Somerville (1999), 129–256. Search in Google Scholar
[29] M. V. Karasëv, Analogues of objects of the theory of Lie groups for nonlinear Poisson brackets, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), no. 3, 508–538, 638. 10.1070/IM1987v028n03ABEH000895Search in Google Scholar
[30] M. V. Karasëv, The Maslov quantization conditions in higher cohomology and analogs of notions developed in Lie theory for canonical fibre bundles of symplectic manifolds. I, II, Selecta Math. Soviet 8 (1989), 213–234, 235–258. Search in Google Scholar
[31] K. Kodaira, Complex manifolds and deformation of complex structures. Translated from the Japanese by Kazuo Akao, With an appendix by Daisuke Fujiwara, Grundlehren Math. Wiss. 283, Springer, New York 1986. 10.1007/978-1-4613-8590-5Search in Google Scholar
[32] A. G. Kovalev, Nahm’s equations and complex adjoint orbits, Quart. J. Math. Oxford Ser. (2) 47 (1996), no. 185, 41–58. 10.1093/qmath/47.1.41Search in Google Scholar
[33] P. B. Kronheimer, A hyperkahler structure on the cotangent bundle of a complex Lie group, preprint (1988), https://arxiv.org/abs/math/0409253. Search in Google Scholar
[34] P. B. Kronheimer, The construction of ALE spaces as hyper-Kähler quotients, J. Differential Geom. 29 (1989), no. 3, 665–683. 10.1142/9789814539395_0034Search in Google Scholar
[35] P. B. Kronheimer, A hyper-Kählerian structure on coadjoint orbits of a semisimple complex group, J. Lond. Math. Soc. (2) 42 (1990), no. 2, 193–208. 10.1112/jlms/s2-42.2.193Search in Google Scholar
[36] P. B. Kronheimer, Instantons and the geometry of the nilpotent variety, J. Differential Geom. 32 (1990), no. 2, 473–490. 10.4310/jdg/1214445316Search in Google Scholar
[37] C. Laurent-Gengoux, M. Stiénon and P. Xu, Holomorphic Poisson manifolds and holomorphic Lie algebroids, Int. Math. Res. Not. IMRN 2008 (2008), Article ID rnn 088. 10.1093/imrn/rnn088Search in Google Scholar
[38] C. Laurent-Gengoux, M. Stiénon and P. Xu, Integration of holomorphic Lie algebroids, Math. Ann. 345 (2009), no. 4, 895–923. 10.1007/s00208-009-0388-7Search in Google Scholar
[39] A. Maciocia, Metrics on the moduli spaces of instantons over Euclidean 4-space, Comm. Math. Phys. 135 (1991), no. 3, 467–482. 10.1007/BF02104116Search in Google Scholar
[40] K. C. H. Mackenzie and P. Xu, Integration of Lie bialgebroids, Topology 39 (2000), no. 3, 445–467. 10.1016/S0040-9383(98)00069-XSearch in Google Scholar
[41] C. B. Morrey, Jr., On the analyticity of the solutions of analytic non-linear elliptic systems of partial differential equations. I. Analyticity in the interior, Amer. J. Math. 80 (1958), 198–218. 10.2307/2372830Search in Google Scholar
[42] H. Nakajima, Instantons on ALE spaces, quiver varieties, and Kac–Moody algebras, Duke Math. J. 76 (1994), no. 2, 365–416. 10.1215/S0012-7094-94-07613-8Search in Google Scholar
[43] A. Newlander and L. Nirenberg, Complex analytic coordinates in almost complex manifolds, Ann. of Math. (2) 65 (1957), 391–404. 10.2307/1970051Search in Google Scholar
[44] R. Penrose, Nonlinear gravitons and curved twistor theory, Gen. Relativity Gravitation 7 (1976), no. 1, 31–52. 10.1007/BF00762011Search in Google Scholar
[45] B. Pym, Constructions and classifications of projective Poisson varieties, Lett. Math. Phys. 108 (2018), no. 3, 573–632. 10.1007/s11005-017-0984-5Search in Google Scholar PubMed PubMed Central
[46] C. Simpson, The Hodge filtration on nonabelian cohomology, Algebraic geometry—Santa Cruz 1995, Proc. Sympos. Pure Math. 62, American Mathematical Society, Providence (1997), 217–281. 10.1090/pspum/062.2/1492538Search in Google Scholar
[47] A. Weinstein, The local structure of Poisson manifolds, J. Differential Geom. 18 (1983), no. 3, 523–557. 10.4310/jdg/1214437787Search in Google Scholar
[48] A. Weinstein, Symplectic groupoids and Poisson manifolds, Bull. Amer. Math. Soc. (N.S.) 16 (1987), no. 1, 101–104. 10.1090/S0273-0979-1987-15473-5Search in Google Scholar
[49] R. O. Wells, Jr., Differential analysis on complex manifolds. With a new appendix by Oscar Garcia-Prada, 3rd ed., Grad. Texts in Math. 65, Springer, New York 2008. 10.1007/978-0-387-73892-5Search in Google Scholar
[50] S. T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation. I, Comm. Pure Appl. Math. 31 (1978), no. 3, 339–411. 10.1002/cpa.3160310304Search in Google Scholar
[51] S. Zakrzewski, Quantum and classical pseudogroups. I. Union pseudogroups and their quantization, Comm. Math. Phys. 134 (1990), no. 2, 347–370. 10.1007/BF02097706Search in Google Scholar
[52] S. Zakrzewski, Quantum and classical pseudogroups. II. Differential and symplectic pseudogroups, Comm. Math. Phys. 134 (1990), no. 2, 371–395. 10.1007/BF02097707Search in Google Scholar
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Articles in the same Issue
- Frontmatter
- On the torsion values for sections of an elliptic scheme
- 2-Verma modules
- Ellipticity and discrete series
- Quasi-isometric rigidity for graphs of virtually free groups with two-ended edge groups
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Articles in the same Issue
- Frontmatter
- On the torsion values for sections of an elliptic scheme
- 2-Verma modules
- Ellipticity and discrete series
- Quasi-isometric rigidity for graphs of virtually free groups with two-ended edge groups
- Nonnegative Ricci curvature and escape rate gap
- Hyperkähler metrics near Lagrangian submanifolds and symplectic groupoids
- On the geometry of lattices and finiteness of Picard groups
- About every convex set in any generic Riemannian manifold
- Hopf-type theorem for self-shrinkers
