Blow-Ups in Generalized Kähler Geometry

We study blow-ups in generalized Kähler geometry. The natural candidates for submanifolds to be blown-up are those which are generalized Poisson submanifolds for one of the two generalized complex structures and can be blown up in a generalized complex manner. We show that the bi-Hermitian structure...

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Bibliographic Details
Published in:Communications in mathematical physics 2018-02, Vol.357 (3), p.1133-1156
Main Author: van der Leer Durán, J. L.
Format: Article
Language:English
Subjects:
Classical and Quantum Gravitation
Complex Systems
Deformation
Geometry
Lie groups
Lifts
Manifolds (mathematics)
Mathematical and Computational Physics
Mathematical Physics
Physics
Physics and Astronomy
Quantum Physics
Relativity Theory
Theoretical
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Summary:We study blow-ups in generalized Kähler geometry. The natural candidates for submanifolds to be blown-up are those which are generalized Poisson submanifolds for one of the two generalized complex structures and can be blown up in a generalized complex manner. We show that the bi-Hermitian structure underlying the generalized Kähler pair lifts to a degenerate bi-Hermitian structure on this blow-up. Then, using a deformation procedure based on potentials in Kähler geometry, we identify two concrete situations in which one can deform the degenerate structure on the blow-up into a non-degenerate one. We end with a study of generalized Kähler Lie groups and give a concrete example on ( S 1 ) n × ( S 3 ) m , for n +  m even.
ISSN:0010-3616
1432-0916
DOI:10.1007/s00220-017-3039-y