An example of compact Kähler manifold with nonnegative quadratic bisectional curvature

Li, Qun; Wu, Damin; Zheng, Fangyang

doi:10.1090/S0002-9939-2013-11596-0

An example of compact Kähler manifold with nonnegative quadratic bisectional curvature
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by Qun Li, Damin Wu and Fangyang Zheng

Proc. Amer. Math. Soc. 141 (2013), 2117-2126

DOI: https://doi.org/10.1090/S0002-9939-2013-11596-0

Published electronically: February 12, 2013

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Abstract:

We construct a compact Kähler manifold of nonnegative quadratic bisectional curvature which does not admit any Kähler metric of nonnegative orthogonal bisectional curvature. The manifold is a 7-dimensional Kähler $C$-space with second Betti number equal to 1, and its canonical metric is a Kähler-Einstein metric of positive scalar curvature.

References

Armand Borel, Kählerian coset spaces of semisimple Lie groups, Proc. Nat. Acad. Sci. U.S.A. 40 (1954), 1147–1151. MR 77878, DOI 10.1073/pnas.40.12.1147
Armand Borel, On the curvature tensor of the Hermitian symmetric manifolds, Ann. of Math. (2) 71 (1960), 508–521. MR 111059, DOI 10.2307/1969940
A. Borel and F. Hirzebruch, Characteristic classes and homogeneous spaces. I, Amer. J. Math. 80 (1958), 458–538. MR 102800, DOI 10.2307/2372795
Simon Brendle and Richard M. Schoen, Classification of manifolds with weakly $1/4$-pinched curvatures, Acta Math. 200 (2008), no. 1, 1–13. MR 2386107, DOI 10.1007/s11511-008-0022-7
Simon Brendle and Richard Schoen, Manifolds with $1/4$-pinched curvature are space forms, J. Amer. Math. Soc. 22 (2009), no. 1, 287–307. MR 2449060, DOI 10.1090/S0894-0347-08-00613-9
A. Chau and L. F. Tam, On quadratic orthogonal bisectional curvature, preprint. To appear in J. Differential Geom.
X. X. Chen, On Kähler manifolds with positive orthogonal bisectional curvature, Adv. Math. 215 (2007), no. 2, 427–445. MR 2355611, DOI 10.1016/j.aim.2006.11.006
Jean-Pierre Demailly, Thomas Peternell, and Michael Schneider, Compact complex manifolds with numerically effective tangent bundles, J. Algebraic Geom. 3 (1994), no. 2, 295–345. MR 1257325
J. X. Fu, Z. Wang, and D. Wu, Form-type equations on Kähler manifolds of nonnegative orthogonal bisectional curvature, arXiv: 1010.2022
HuiLing Gu and ZhuHong Zhang, An extension of Mok’s theorem on the generalized Frankel conjecture, Sci. China Math. 53 (2010), no. 5, 1253–1264. MR 2653275, DOI 10.1007/s11425-010-0013-y
Alan Howard, Brian Smyth, and H. Wu, On compact Kähler manifolds of nonnegative bisectional curvature. I, Acta Math. 147 (1981), no. 1-2, 51–56. MR 631087, DOI 10.1007/BF02392867
Mitsuhiro Itoh, On curvature properties of Kähler $C$-spaces, J. Math. Soc. Japan 30 (1978), no. 1, 39–71. MR 470904, DOI 10.2969/jmsj/03010039
Q. Li, D. Wu, and F. Zheng, Quadratic bisectional curvature and constant rank theorems, in preparation.
Ngaiming Mok, The uniformization theorem for compact Kähler manifolds of nonnegative holomorphic bisectional curvature, J. Differential Geom. 27 (1988), no. 2, 179–214. MR 925119
Hsien-Chung Wang, Closed manifolds with homogeneous complex structure, Amer. J. Math. 76 (1954), 1–32. MR 66011, DOI 10.2307/2372397
H. Wu, On compact Kähler manifolds of nonnegative bisectional curvature. II, Acta Math. 147 (1981), no. 1-2, 57–70. MR 631088, DOI 10.1007/BF02392868
Damin Wu, Shing-Tung Yau, and Fangyang Zheng, A degenerate Monge-Ampère equation and the boundary classes of Kähler cones, Math. Res. Lett. 16 (2009), no. 2, 365–374. MR 2496750, DOI 10.4310/MRL.2009.v16.n2.a12
Xiangwen Zhang, On the boundary of Kähler cones, Proc. Amer. Math. Soc. 140 (2012), no. 2, 701–705. MR 2846339, DOI 10.1090/S0002-9939-2011-10929-8
Fangyang Zheng, Complex differential geometry, AMS/IP Studies in Advanced Mathematics, vol. 18, American Mathematical Society, Providence, RI; International Press, Boston, MA, 2000. MR 1777835, DOI 10.1090/amsip/018