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Hermitian metrics of positive holomorphic sectional curvature on fibrations

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Abstract

The main result of this note essentially is that if the base and fibers of a compact fibration carry Hermitian metrics of positive holomorphic sectional curvature, then so does the total space of the fibration. The proof is based on the use of a warped product metric as in the work by Cheung in case of negative holomorphic sectional curvature, but differs in certain key aspects, e.g., in that it does not use the subadditivity property for holomorphic sectional curvature due to Grauert-Reckziegel and Wu.

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References

  1. Alvarez, A., Chaturvedi, A., Heier, G.: Optimal pinching for the holomorphic sectional curvature of Hitchin’s metrics on Hirzebruch surfaces. Contemp. Math. 654, 133–142 (2015)

    Article  MathSciNet  Google Scholar 

  2. Alvarez, A., Heier, G., Zheng, F.: On projectivized vector bundles and positive holomorphic sectional curvature. Proc. Am. Math. Soc. 146(7), 2877–2882 (2018)

    Article  MathSciNet  Google Scholar 

  3. Bishop, R., O’Neill, B.: Manifolds of negative curvature. Trans. Am. Math. Soc. 145, 1–49 (1969)

    Article  MathSciNet  Google Scholar 

  4. Chaturvedi, A.: On holomorphic sectional curvature and fibrations. Ph.D. Thesis, University of Houston (2016)

  5. Cheung, C.-K.: Hermitian metrics of negative holomorphic sectional curvature on some hyperbolic manifolds. Math. Z. 201(1), 105–119 (1989)

    Article  MathSciNet  Google Scholar 

  6. Cowen, M.: Families of negatively curved Hermitian manifolds. Proc. Am. Math. Soc. 39, 362–366 (1973)

    Article  MathSciNet  Google Scholar 

  7. Diverio, S., Trapani, S.: Quasi-negative holomorphic sectional curvature and positivity of the canonical bundle. J. Differ. Geom. 111(2), 303–314 (2019)

    Article  MathSciNet  Google Scholar 

  8. Graber, T., Harris, J., Starr, J.: Families of rationally connected varieties. J. Am. Math. Soc. 16(1), 57–67 (2003). (electronic)

    Article  MathSciNet  Google Scholar 

  9. Grauert, H., Reckziegel, H.: Hermitesche metriken und normale familien holomorpher abbildungen. Math. Z. 89, 108–125 (1965)

    Article  MathSciNet  Google Scholar 

  10. Hitchin, N.: On the curvature of rational surfaces. In: Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Part 2, Stanford Univ., Stanford, Calif., 1973), pages 65–80. Am. Math. Soc., Providence, R. I. (1975)

  11. Heier, G., Lu, S., Wong, B.: On the canonical line bundle and negative holomorphic sectional curvature. Math. Res. Lett. 17(6), 1101–1110 (2010)

    Article  MathSciNet  Google Scholar 

  12. Heier, G., Lu, S., Wong, B.: Kähler manifolds of semi-negative holomorphic sectional curvature. J. Differ. Geom. 104(3), 419–441 (2016)

    Article  Google Scholar 

  13. Heier, G., Lu, S., Wong, B., Zheng, F.: Reduction of manifolds with semi-negative holomorphic sectional curvature. Math. Ann. 372(3–4), 951–962 (2018)

    Article  MathSciNet  Google Scholar 

  14. Heier, G., Wong, B.: Scalar curvature and uniruledness on projective manifolds. Comm. Anal. Geom. 20(4), 751–764 (2012)

    Article  MathSciNet  Google Scholar 

  15. Heier, G., Wong, B.: On projective Kähler manifolds of partially positive curvature and rational connectedness. arXiv:1509.02149v1 (2015)

  16. Tsukamoto, Y.: On Kählerian manifolds with positive holomorphic sectional curvature. Proc. Jpn. Acad. 33, 333–335 (1957)

    Article  Google Scholar 

  17. Tosatti, V., Yang, X.: An extension of a theorem of Wu-Yau. J. Differ. Geom. 107(3), 573–579 (2017)

    Article  MathSciNet  Google Scholar 

  18. Wu, H.-H.: A remark on holomorphic sectional curvature. Indiana Univ. Math. J. 22, 1103–1108 (1973)

    Article  MathSciNet  Google Scholar 

  19. Wu, D., Yau, S.-T.: Negative holomorphic curvature and positive canonical bundle. Invent. Math. 204(2), 595–604 (2016)

    Article  MathSciNet  Google Scholar 

  20. Wu, D., Yau, S.-T.: A remark on our paper “Negative holomorphic curvature and positive canonical bundle”. Comm. Anal. Geom. 24(4), 901–912 (2016)

    Article  MathSciNet  Google Scholar 

  21. Yang, X.: Hermitian manifolds with semi-positive holomorphic sectional curvature. Math. Res. Lett. 23(3), 939–952 (2016)

    Article  MathSciNet  Google Scholar 

  22. Yang, B., Zheng, F.: Hirzebruch manifolds and positive holomorphic sectional curvature. arXiv:1611.06571v2 (2016)

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Acknowledgements

This work was part of the first author’s Ph.D. thesis written under the direction of the second author at the University of Houston. The first author would like to thank TIFR Center for Applicable Mathematics, Bengaluru and TIFR, Mumbai for support during the time this paper was finalized.

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Authors and Affiliations

  1. Department of Mathematics, Tata Institute of Fundamental Research, Dr Homi Bhabha Road, Navy Nagar, Colaba, Mumbai, Maharashtra, 400005, India

    Ananya Chaturvedi

  2. Department of Mathematics, University of Houston, 4800 Calhoun Road, Houston, TX, 77204, USA

    Gordon Heier

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  1. Ananya Chaturvedi
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  2. Gordon Heier
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Correspondence to Gordon Heier.

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Chaturvedi, A., Heier, G. Hermitian metrics of positive holomorphic sectional curvature on fibrations. Math. Z. 295, 349–364 (2020). https://doi.org/10.1007/s00209-019-02341-6

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  • DOI: https://doi.org/10.1007/s00209-019-02341-6

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