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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Blow-ups in generalized complex geometry


Authors: M. A. Bailey, G. R. Cavalcanti and J. L. van der Leer Durán
Journal: Trans. Amer. Math. Soc. 371 (2019), 2109-2131
MSC (2010): Primary 53D18
DOI: https://doi.org/10.1090/tran/7412
Published electronically: October 1, 2018
MathSciNet review: 3894047
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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.


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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
Email: michael.bailey.math@gmail.com

G. R. Cavalcanti
Affiliation: Department of Mathematics, Utrecht University, 3584 CD Utrecht, The Netherlands
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

DOI: https://doi.org/10.1090/tran/7412
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.
Article copyright: © Copyright 2018 American Mathematical Society