Abstract
In this paper, we show that any compact Kähler manifold homotopic to a compact Riemannian manifold with negative sectional curvature admits a Kähler–Einstein metric of general type. Moreover, we prove that, on a compact symplectic manifold X homotopic to a compact Riemannian manifold with negative sectional curvature, for any almost complex structure J compatible with the symplectic form, there is no non-constant J-holomorphic entire curve \(f:{\mathbb C \,}\rightarrow X\).
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Acknowledgements
B.-L. Chen was partially supported by Grants NSFC11521101, 11025107. X.-K. Yang was partially supported by China’s Recruitment Program of Global Experts and National Center for Mathematics and Interdisciplinary Sciences, Chinese Academy of Sciences. The authors would like to thank the anonymous referee for pointing out a similar proof in Lemma 3.2.
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Communicated by Ngaiming Mok.
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Chen, BL., Yang, X. Compact Kähler manifolds homotopic to negatively curved Riemannian manifolds. Math. Ann. 370, 1477–1489 (2018). https://doi.org/10.1007/s00208-017-1521-7
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DOI: https://doi.org/10.1007/s00208-017-1521-7