Blow-ups in generalized complex geometry
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- by M. A. Bailey, G. R. Cavalcanti and J. L. van der Leer Durán PDF
- Trans. Amer. Math. Soc. 371 (2019), 2109-2131 Request permission
Abstract:
We study blow-ups in generalized complex geometry. To that end we introduce the concept of holomorphic ideals, which allows one to define a blow-up in the category of smooth manifolds. We then investigate which generalized complex submanifolds are suitable for blowing up. Two classes naturally appear: generalized Poisson submanifolds and generalized Poisson transversals. These are submanifolds for which the geometry normal to the submanifold is complex, respectively symplectic. We show that generalized Poisson submanifolds carry a canonical holomorphic ideal, and we give a necessary and sufficient condition for the corresponding blow-up to be generalized complex. For generalized Poisson transversals we prove a normal form theorem for a neighborhood of the submanifold and use it to define a generalized complex blow-up.References
- Michael F. Atiyah, Vector fields on manifolds, Arbeitsgemeinschaft für Forschung des Landes Nordrhein-Westfalen, Heft 200, Westdeutscher Verlag, Cologne, 1970 (English, with German and French summaries). MR 0263102
- Michael Bailey, Local classification of generalized complex structures, J. Differential Geom. 95 (2013), no. 1, 1–37. MR 3128977
- Michael Bailey and Marco Gualtieri, Local analytic geometry of generalized complex structures, Bull. Lond. Math. Soc. 49 (2017), no. 2, 307–319. MR 3656299, DOI 10.1112/blms.12029
- W. Barth, C. Peters, and A. Van de Ven, Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 4, Springer-Verlag, Berlin, 1984.
- Henrique Bursztyn, A brief introduction to Dirac manifolds, Geometric and topological methods for quantum field theory, Cambridge Univ. Press, Cambridge, 2013, pp. 4–38. MR 3098084
- Henrique Bursztyn, Gil R. Cavalcanti, and Marco Gualtieri, Reduction of Courant algebroids and generalized complex structures, Adv. Math. 211 (2007), no. 2, 726–765. MR 2323543, DOI 10.1016/j.aim.2006.09.008
- Gil R. Cavalcanti and Marco Gualtieri, Blow-up of generalized complex 4-manifolds, J. Topol. 2 (2009), no. 4, 840–864. MR 2574746, DOI 10.1112/jtopol/jtp031
- Pedro Frejlich and Ioan Mărcuţ, The normal form theorem around Poisson transversals, Pacific J. Math. 287 (2017), no. 2, 371–391. MR 3632892, DOI 10.2140/pjm.2017.287.371
- Mikhael Gromov, Partial differential relations, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 9, Springer-Verlag, Berlin, 1986. MR 864505, DOI 10.1007/978-3-662-02267-2
- Marco Gualtieri, Generalized complex geometry, Ann. of Math. (2) 174 (2011), no. 1, 75–123. MR 2811595, DOI 10.4007/annals.2011.174.1.3
- Marco Gualtieri, Generalized Kähler geometry, Comm. Math. Phys. 331 (2014), no. 1, 297–331. MR 3232003, DOI 10.1007/s00220-014-1926-z
- Heinz Hopf, Schlichte Abbildungen und lokale Modifikationen $4$-dimensionaler komplexer Mannigfaltigkeiten, Comment. Math. Helv. 29 (1955), 132–156 (German). MR 68008, DOI 10.1007/BF02564276
- Eugene Lerman, Symplectic cuts, Math. Res. Lett. 2 (1995), no. 3, 247–258. MR 1338784, DOI 10.4310/MRL.1995.v2.n3.a2
- B. Malgrange, Ideals of differentiable functions, Tata Institute of Fundamental Research Studies in Mathematics, vol. 3, Tata Institute of Fundamental Research, Bombay; Oxford University Press, London, 1967. MR 0212575
- Dusa McDuff, Examples of simply-connected symplectic non-Kählerian manifolds, J. Differential Geom. 20 (1984), no. 1, 267–277. MR 772133
- A. Polishchuk, Algebraic geometry of Poisson brackets, J. Math. Sci. (New York) 84 (1997), no. 5, 1413–1444. Algebraic geometry, 7. MR 1465521, DOI 10.1007/BF02399197
- J. L. van der Leer Durán, Blow-ups in generalized Kähler geometry, Comm. Math. Phys. 357 (2018), no. 3, 1133–1156. MR 3769747, DOI 10.1007/s00220-017-3039-y
- Oscar Zariski, Normal varieties and birational correspondences, Bull. Amer. Math. Soc. 48 (1942), 402–413. MR 6451, DOI 10.1090/S0002-9904-1942-07685-3
Additional Information
- M. A. Bailey
- Affiliation: Mathematics Institute, Utrecht University, 3508 TA Utrecht, The Netherlands
- Address at time of publication: Department of Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
- MR Author ID: 1043462
- Email: michael.bailey.math@gmail.com
- G. R. Cavalcanti
- Affiliation: Department of Mathematics, Utrecht University, 3584 CD Utrecht, The Netherlands
- MR Author ID: 757552
- Email: gil.cavalcanti@gmail.com
- J. L. van der Leer Durán
- Affiliation: Department of Mathematics, Utrecht University, 3584 CD Utrecht, The Netherlands
- Address at time of publication: Department of Mathematics, University of Toronto, Toronto, Ontario M55 2E4, Canada
- Email: joeyvdld@gmail.com
- Received by editor(s): July 22, 2016
- Received by editor(s) in revised form: September 22, 2017
- Published electronically: October 1, 2018
- Additional Notes: The first and second authors were supported by the VIDI grant 639.032.221.
The third author was supported by the Free Competition Grant 613.001.112 from NWO, the Netherlands Organisation for Scientific Research. - © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 2109-2131
- MSC (2010): Primary 53D18
- DOI: https://doi.org/10.1090/tran/7412
- MathSciNet review: 3894047