Kähler-Ricci flow on projective bundles over Kähler-Einstein manifolds
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- by Frederick Tsz-Ho Fong PDF
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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.References
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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
- 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.
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 366 (2014), 563-589
- MSC (2010): Primary 53C44, 53C55; Secondary 55R25
- DOI: https://doi.org/10.1090/S0002-9947-2013-05726-1
- MathSciNet review: 3130308