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

 

Kähler-Ricci flow on projective bundles over Kähler-Einstein manifolds


Author: Frederick Tsz-Ho Fong
Journal: Trans. Amer. Math. Soc. 366 (2014), 563-589
MSC (2010): Primary 53C44, 53C55; Secondary 55R25
Published electronically: August 14, 2013
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Abstract: We study the Kähler-Ricci flow on a class of projective bundles $ \mathbb{P}(\mathcal {O}_\Sigma \oplus L)$ over the compact Kähler-Einstein manifold $ \Sigma ^n$. Assuming the initial Kähler metric $ \omega _0$ admits a $ U(1)$-invariant momentum profile, we give a criterion, characterized by the triple $ (\Sigma , L, [\omega _0])$, under which the $ \mathbb{P}^1$-fiber collapses along the Kähler-Ricci flow and the projective bundle converges to $ \Sigma $ in the Gromov-Hausdorff sense. Furthermore, the Kähler-Ricci flow must have Type I singularity and is of $ (\mathbb{C}^n \times \mathbb{P}^1)$-type. This generalizes and extends part of Song-Weinkove's work on Hirzebruch surfaces.


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Additional Information

Frederick Tsz-Ho Fong
Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305
Address at time of publication: Department of Mathematics, Brown University, 151 Thayer Street, Box 1917, Providence, Rhode Island 02912
Email: thfong@math.stanford.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05726-1
PII: S 0002-9947(2013)05726-1
Keywords: K\"ahler-Ricci flow, singularity analysis, projective bundles
Received by editor(s): April 13, 2011
Received by editor(s) in revised form: September 30, 2011, and October 12, 2011
Published electronically: August 14, 2013
Additional Notes: The author was supported in part by NSF Grant DMS-#0604960.
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.