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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Deformation of Sasakian metrics


Author: Hiraku Nozawa
Journal: Trans. Amer. Math. Soc. 366 (2014), 2737-2771
MSC (2010): Primary 32G07; Secondary 53C25
Published electronically: November 5, 2013
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Abstract: Deformations of the Reeb flow of a Sasakian manifold as transversely Kähler flows may not admit compatible Sasakian metrics. We show that the triviality of the $ (0,2)$-component of the basic Euler class characterizes the existence of compatible Sasakian metrics for given small deformations of the Reeb flow as transversely holomorphic Riemannian flows. We also prove a Kodaira-Akizuki-Nakano type vanishing theorem for basic Dolbeault cohomology of homologically orientable transversely Kähler foliations. As a consequence of these results, we show that any small deformations of the Reeb flow of a positive Sasakian manifold admit compatible Sasakian metrics.


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Additional Information

Hiraku Nozawa
Affiliation: Institut des Hautes Études Scientifiques, Le Bois-Marie 35, Route de Chartres 91440 Bures-sur-Yvette, France
Address at time of publication: Department of Mathematical Sciences, Faculty of Science and Engineering, Ritsumeikan University, Nojihigashi 1-1-1, Kusatsu, Shiga, 526-8755, Japan
Email: nozawahiraku@06.alumni.u-tokyo.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-06020-5
PII: S 0002-9947(2013)06020-5
Keywords: Deformation theory, Sasakian metrics, transversely holomorphic foliations
Received by editor(s): September 15, 2010
Received by editor(s) in revised form: October 6, 2011, and October 3, 2012
Published electronically: November 5, 2013
Additional Notes: The author was partially supported by Grant-in-Aid for JSPS Fellows (19-4609) and Postdoctoral Fellowship of French government (662014L)
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.