Abstract
We show that any proper Lie groupoid admits a compatible real analytic structure. Our proof hinges on a Weyl unitary trick of sorts for appropriate local holomorphic groupoids.
Acknowledgements
The author is grateful to P. Frejlich and G. Trentinaglia for useful discussions on the subject. He would also like to thank the referee for a most thorough and acute report.
References
[1] R. M. Aguilar, Symplectic reduction and the homogeneous complex Monge–Ampère equation, Ann. Global Anal. Geom. 19 (2001), no. 4, 327–353. 10.1023/A:1010715415333Search in Google Scholar
[2] R. H. Bing, A homeomorphism between the 3-sphere and the sum of two solid horned spheres, Ann. of Math. (2) 56 (1952), 354–362. 10.2307/1969804Search in Google Scholar
[3] A. Cabrera, I. Mărcuţ and M. A. Salazar, On local integration of Lie brackets, J. reine angew. Math. 760 (2020), 267–293. 10.1515/crelle-2018-0011Search in Google Scholar
[4] M. Crainic and R. L. Fernandes, Lectures on integrability of Lie brackets, Lectures on Poisson geometry, Geom. Topol. Monogr. 17, Geometry & Topology, Coventry (2011), 1–107. Search in Google Scholar
[5] M. Crainic, R. L. Fernandes and D. Martínez Torres, Poisson manifolds of compact types (PMCT 1), J. reine angew. Math. 756 (2019), 101–149. 10.1515/crelle-2017-0006Search in Google Scholar
[6] M. Crainic, R. L. Fernandes and D. Martínez Torres, Regular Poisson manifolds of compact types, Astérisque 413, Société Mathématique de France, Paris 2019. 10.24033/ast.1079Search in Google Scholar
[7] M. Crainic and J. A. N. Mestre, Measures on differentiable stacks, J. Noncommut. Geom. 13 (2019), no. 4, 1235–1294. 10.4171/JNCG/362Search in Google Scholar
[8] M. Crainic and I. Struchiner, On the linearization theorem for proper Lie groupoids, Ann. Sci. Éc. Norm. Supér. (4) 46 (2013), no. 5, 723–746. 10.24033/asens.2200Search in Google Scholar
[9] M. del Hoyo and R. L. Fernandes, Riemannian metrics on Lie groupoids, J. reine angew. Math. 735 (2018), 143–173. 10.1515/crelle-2015-0018Search in Google Scholar
[10] M. L. del Hoyo, Lie groupoids and their orbispaces, Port. Math. 70 (2013), no. 2, 161–209. 10.4171/PM/1930Search in Google Scholar
[11] R. Edwards, K. Millett and D. Sullivan, Foliations with all leaves compact, Topology 16 (1977), no. 1, 13–32. 10.1016/0040-9383(77)90028-3Search in Google Scholar
[12] D. B. A. Epstein, Foliations with all leaves compact, Ann. Inst. Fourier (Grenoble) 26 (1976), no. 1, 265–282. 10.1007/BFb0082573Search in Google Scholar
[13] R. L. Fernandes and D. Michiels, Associativity and integrability, preprint (2018), https://arxiv.org/abs/1803.10402. 10.1090/tran/8073Search in Google Scholar
[14] A. M. Gleason, Groups without small subgroups, Ann. of Math. (2) 56 (1952), 193–212. 10.2307/1969795Search in Google Scholar
[15] I. Goldbring, Hilbert’s fifth problem for local groups, Ann. of Math. (2) 172 (2010), no. 2, 1269–1314. 10.4007/annals.2010.172.1269Search in Google Scholar
[16] K. Grove, H. Karcher and E. A. Ruh, Group actions and curvature, Bull. Amer. Math. Soc. 81 (1975), 89–92. 10.1090/S0002-9904-1975-13647-0Search in Google Scholar
[17] V. Guillemin and M. Stenzel, Grauert tubes and the homogeneous Monge–Ampère equation, J. Differential Geom. 34 (1991), no. 2, 561–570. 10.4310/jdg/1214447221Search in Google Scholar
[18] P. Heinzner, Geometric invariant theory on Stein spaces, Math. Ann. 289 (1991), no. 4, 631–662. 10.1007/BF01446594Search in Google Scholar
[19] S. Illman, Every proper smooth action of a Lie group is equivalent to a real analytic action: A contribution to Hilbert’s fifth problem, Prospects in topology (Princeton 1994), Ann. of Math. Stud. 138, Princeton University, Princeton (1995), 189–220. 10.1515/9781400882588-014Search in Google Scholar
[20] S. Illman, Group actions and Hilbert’s fifth problem, Handbook of global analysis, Elsevier Science, Amsterdam (2008), 533–589, 1213. 10.1016/B978-044452833-9.50011-5Search in Google Scholar
[21] F. Kutzschebauch and F. Loose, Real analytic structures on a symplectic manifold, Proc. Amer. Math. Soc. 128 (2000), no. 10, 3009–3016. 10.1090/S0002-9939-00-05713-0Search in Google Scholar
[22] L. Lempert and R. Szőke, Global solutions of the homogeneous complex Monge–Ampère equation and complex structures on the tangent bundle of Riemannian manifolds, Math. Ann. 290 (1991), no. 4, 689–712. 10.1007/BF01459268Search in Google Scholar
[23] I. Moerdijk and J. Mrčun, Introduction to foliations and Lie groupoids, Cambridge Stud. Adv. Math. 91, Cambridge University, Cambridge 2003. 10.1017/CBO9780511615450Search in Google Scholar
[24] D. Montgomery and L. Zippin, Small subgroups of finite-dimensional groups, Ann. of Math. (2) 56 (1952), 213–241. 10.2307/1969796Search in Google Scholar
[25] D. Montgomery and L. Zippin, Examples of transformation groups, Proc. Amer. Math. Soc. 5 (1954), 460–465. 10.1090/S0002-9939-1954-0062436-2Search in Google Scholar
[26] R. Narasimhan, Analysis on real and complex manifolds, Adv. Stud. Pure Math. 1, Masson & Cie, Paris 1968. Search in Google Scholar
[27] J. Pardon, The Hilbert–Smith conjecture for three-manifolds, J. Amer. Math. Soc. 26 (2013), no. 3, 879–899. 10.1090/S0894-0347-2013-00766-3Search in Google Scholar
[28] T. Tao, Hilbert’s fifth problem and related topics, Grad. Stud. Math. 153, American Mathematical Society, Providence 2014. 10.1090/gsm/153Search in Google Scholar
[29] N. T. Zung, Proper groupoids and momentum maps: Linearization, affinity, and convexity, Ann. Sci. Éc. Norm. Supér. (4) 39 (2006), no. 5, 841–869. 10.1016/j.ansens.2006.09.002Search in Google Scholar
© 2020 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Restriction formula and subadditivity property related to multiplier ideal sheaves
- Proper Lie groupoids are real analytic
- Deformations of rational curves in positive characteristic
- Curved Rickard complexes and link homologies
- Boundary properties of fractional objects: Flexibility of linear equations and rigidity of minimal graphs
Articles in the same Issue
- Frontmatter
- Restriction formula and subadditivity property related to multiplier ideal sheaves
- Proper Lie groupoids are real analytic
- Deformations of rational curves in positive characteristic
- Curved Rickard complexes and link homologies
- Boundary properties of fractional objects: Flexibility of linear equations and rigidity of minimal graphs