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Complete Kähler metrics with positive holomorphic sectional curvatures on certain line bundles (related to a cohomogeneity one point of view on a Yau conjecture)

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Abstract

In this article, we study Kähler metrics on a certain line bundle over some compact Kähler manifolds to find complete Kähler metrics with positive holomorphic sectional (or bisectional) curvatures. Thus, we apply a strategy to a famous Yau conjecture with a co-homogeneity one geometry.

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References

  1. Calabi E. Extremal Kähler metrics//Yau S T. Seminars on Differential Geometry. Princeton: Princeton University Press, 1982: 259–290

    Google Scholar 

  2. Calabi E. Extremal Kähler metrics II//Chavel I, Farkas H M. Differential Geometry and Complex Analysis. Berlin: Springer-Verlag, 1985: 95–114

    Chapter  Google Scholar 

  3. Chen B, Zhu X. Yau’s uniformization conjecture for manifolds with non-maximal volume growth. Acta Mathematica Scientia, 2018, 38B: 1468–1484

    Article  MathSciNet  MATH  Google Scholar 

  4. Demailly J, Peternell T, Schneider M. Compact complex manifolds with numerically effective tangent bundles. J Alg Geom, 1994, 3: 295–345

    MathSciNet  MATH  Google Scholar 

  5. Duan X M. Complete Kähler Metrics with Positive Holomorphic Sectional Curvatures on Certain Line Bundles [D]. Kaifeng: Henan University, 2022

    Google Scholar 

  6. Guan Z. On Certain Complex Manifolds [D]. Berkeley: University of California, 1993

    Google Scholar 

  7. Guan Z. Existence of extremal metrices on compact almost homogeneous manifolds with two ends. Tran Amer Math Soc, 1995, 347: 2255–2262

    MATH  Google Scholar 

  8. Guan Z. Quasi-einstein metrics. International J Math, 1995, 6(3): 371–379

    Article  MathSciNet  MATH  Google Scholar 

  9. Guan D. On modified Mabuchi functional and Mabuchi moduli space of Kähler metrics on toric bundles. Math Res Letters, 1999, 6: 547–555

    Article  MathSciNet  MATH  Google Scholar 

  10. Guan D. Existence of extremal metrics on almost homogeneous manifolds of cohomogeneity one—III. Intern J Math, 2003, 14: 259–287

    Article  MathSciNet  MATH  Google Scholar 

  11. Guan D. Maxwell-Einstein metrics on certain completion of some C* bundles. Acta Mathematica Scientia, 2023, 43B: 363–372

    Article  MATH  Google Scholar 

  12. Guan D, Liang M. Complete Kähler metrics with positive holomorphic sectional curvatures on certain line bundles (retated to a cohomogeneity one point of view on a Yau Conjecture) II. preprint 2022 and submitted

  13. Greene R, Wu H. Some function-theoretic properties of noncompact Kähler manifolds//Chern S S, Osserman R. Differential Geometry. Providence, RI: Amer Math Soc, 1975: 33–41

    Chapter  Google Scholar 

  14. Greene R, Wu H. C∞ convex functions and manifolds of positive curvature. Acta Math, 1976, 137: 209–245

    Article  MathSciNet  MATH  Google Scholar 

  15. Greene R, Wu H. Analysis on Non-compact Kähler Manifolds//Wells R. Complex Variables. Providence, RI: Amer Math Soc, 1977: 69–102

    Google Scholar 

  16. Greene R, Wu H. On Kähler manifolds of positive bi-sectional curvature and a theorem of Hartogs. Abh Math Sem Univ Hamburg, 1978, 47: 171–185

    Article  MathSciNet  MATH  Google Scholar 

  17. Klembeck P F. A complete Kähler metric of positive curvature on Cn. Proc Amer Math Soc, 1977, 64(2): 313–316

    MathSciNet  MATH  Google Scholar 

  18. Liu G. On Yau’s uniformization conjecture. Cambridge Journal of Mathematics, 2019, 7: 33–70

    Article  MathSciNet  MATH  Google Scholar 

  19. Kobayashi S. Differential Geometry of Complex Vector Bundles. Princeton: Princeton University Press, 1987

    Book  MATH  Google Scholar 

  20. Mok N. The uniformization theorem for compact Kähler manifolds of nonnegative holomorphic bisectional curvature. J Diff Geom, 1988, 27: 179–214

    MATH  Google Scholar 

  21. Mok N. An embedding theorem of complete Kähler manifolds of positive bisectional curvature onto affine algebraic varieties. Bull Soc Math France, 1984, 112: 197–258

    Article  MathSciNet  MATH  Google Scholar 

  22. Mori S. Projective manifolds with ample tangent bundles. Ann Math, 1979, 110: 593–606

    Article  MathSciNet  MATH  Google Scholar 

  23. Mok N, Siu Y T, Yau S T. The Poincaré-Lelong equation on complete Kähler manifolds. Compositio Math, 1981, 44: 183–218

    MathSciNet  MATH  Google Scholar 

  24. Mok N, Zhong J Q. Curvature characterization of compact Hermitian symmetric spaces. J Diff Geom, 1986, 23(1): 15–67

    MathSciNet  MATH  Google Scholar 

  25. Siu Y T. Pseudoconvexity and the problem of levi. Bull Amer Math Soc, 1978, 84: 481–512

    Article  MathSciNet  MATH  Google Scholar 

  26. Siu Y T, Yau S T. Compact Kähler manifolds of positive bisectional curvature. Invent Math, 1980, 59: 189–204

    Article  MathSciNet  MATH  Google Scholar 

  27. Wu H. An elementary method in the study of nonnegative curvature. Acta Math, 1979, 142: 57–78

    Article  MathSciNet  MATH  Google Scholar 

  28. Wu H. Function theory on noncompact Kähler manifolds//Kobayashi S, Horst C, Wu H. Complex Differential Geometry, DMV Sem, 3. Basel: Birkhauser, 1983: 67–155

    Chapter  Google Scholar 

  29. Wu H, Zheng F. Examples of positively curved complete Kähler manifolds//Geometry and Analysis no.1. Adv Lec Math 17. Somerville, MA: Int Press, 2011: 517–542

    MATH  Google Scholar 

  30. Yau S T. Problem Section, Seminar on Differential Geometry. Annals of Math Studies, Vol 102. Princeton: Princeton University Press and University of Tokyo Press, 1982

    MATH  Google Scholar 

  31. Yau S T. A review of complex differential geometry//Proc Symp Pure Math, Vol 52, Part II. Amer Math Soc, 1991

  32. Yang B, Zheng F. Hirzebruch manifolds and positive holomorphic sectional curvature. Ann Inst Fourier (Grenoble), 2019, 69(6): 2589–2634

    Article  MathSciNet  MATH  Google Scholar 

  33. Yang B, Zheng F. Examples of completed Kähler metrics with nonnegative holomorphic sectional curvature. J Geom Anal, 2023, 23: Art 47

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Acknowledgements

We would like to thank Professors Feng, Han and the School of Mathematics and Statistics, Henan University for their support. The second author thanks Professor Peter Li for giving a talk related to this topic when the second author was at UC Riverside, and for his support with [7]. He also thanks Professor X. Zhu for giving a talk at a conference in China, which led him to this problem. Thanks also go to Professor B. Chen for his encouragement. We also take this opportunity to thank Professor Francis Zheng for sending us his paper [29], which is not easy to acquire in China. We also appreciate the constructive comments and suggestions from the referees, which made this paper much more readable.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Henan University, Kaifeng, 475004, China

    Xiaoman Duan & Zhuangdan Guan

  2. Department of Mathematics, The University of California at Riverside, Riverside, CA, 92521, USA

    Zhuangdan Guan

Authors
  1. Xiaoman Duan
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  2. Zhuangdan Guan
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Correspondence to Zhuangdan Guan.

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Conflict of Interest The authors declare no conflict of interest.

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This research was supported by the NSFC (12171140).

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Duan, X., Guan, Z. Complete Kähler metrics with positive holomorphic sectional curvatures on certain line bundles (related to a cohomogeneity one point of view on a Yau conjecture). Acta Math Sci 44, 78–102 (2024). https://doi.org/10.1007/s10473-024-0103-5

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  • DOI: https://doi.org/10.1007/s10473-024-0103-5

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