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

 

On the simply connectedness of nonnegatively curved Kähler manifolds and applications


Authors: Albert Chau and Luen-Fai Tam
Journal: Trans. Amer. Math. Soc. 363 (2011), 6291-6308
MSC (2010): Primary 53C44; Secondary 53C55, 58J37
Published electronically: July 11, 2011
MathSciNet review: 2833555
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Abstract: We study complete noncompact long-time solutions $ (M, g(t))$ to the Kähler-Ricci flow with uniformly bounded nonnegative holomorphic bisectional curvature. We will show that when the Ricci curvature is positive and uniformly pinched, i.e. $ R_{i\bar{\jmath}} \ge cRg_{i\bar{\jmath}}$ at $ (p,t)$ for all $ t$ for some $ c>0$, then there always exists a local gradient Kähler-Ricci soliton limit around $ p$ after possibly rescaling $ g(t)$ along some sequence $ t_i \to \infty$. We will show as an immediate corollary that the injectivity radius of $ g(t)$ along $ t_i$ is uniformly bounded from below along $ t_i$, and thus $ M$ must in fact be simply connected. Additional results concerning the uniformization of $ M$ and fixed points of the holomorphic isometry group will also be established. We will then consider removing the condition of positive Ricci curvature for $ (M, g(t))$. Combining our results with Cao's splitting for Kähler-Ricci flow (2004) and techniques of Ni and Tam (2003), we show that when the positive eigenvalues of the Ricci curvature are uniformly pinched at some point $ p \in M$, then $ M$ has a special holomorphic fiber bundle structure. We will treat as special cases, complete Kähler manifolds with nonnegative holomorphic bisectional curvature and average quadratic curvature decay as well as the case of steady gradient Kähler-Ricci solitons.


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Additional Information

Albert Chau
Affiliation: Department of Mathematics, The University of British Columbia, Room 121, 1984 Mathematics Road, Vancouver, British Columbia, Canada V6T 1Z2
Email: chau@math.ubc.ca

Luen-Fai Tam
Affiliation: The Institute of Mathematical Sciences and Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong, China
Email: lftam@math.cuhk.edu.hk

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05223-2
PII: S 0002-9947(2011)05223-2
Received by editor(s): October 12, 2009
Published electronically: July 11, 2011
Additional Notes: The first author’s research was partially supported by NSERC grant no. #327637-06
The second author’s research was partially supported by Earmarked Grant of Hong Kong #CUHK403108
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.