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On the positivity of high-degree Schur classes of an ample vector bundle

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Abstract

Let X be a smooth projective variety of dimension n, and let E be an ample vector bundle over X. We show that any Schur class of E, lying in the cohomology group of bidegree (n − 1, n − 1), has a representative which is strictly positive in the sense of smooth forms. This conforms the prediction of Griffiths conjecture on the positive polynomials of Chern classes/forms of an ample vector bundle on the form level, and thus strengthens the celebrated positivity results of Fulton and Lazarsfeld (1983) for certain degrees.

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References

  1. Berndtsson B. Curvature of vector bundles associated to holomorphic fibrations. Ann of Math (2), 2009, 169: 531–560

    Article  MathSciNet  Google Scholar 

  2. Biswas I, Pingali V P. Metric properties of parabolic ample bundles. Int Math Res Not IMRN, 2020, 2020: 9336–9369

    Article  MathSciNet  Google Scholar 

  3. Bloch S, Gieseker D. The positivity of the Chern classes of an ample vector bundle. Invent Math, 1971, 12: 112–117

    Article  MathSciNet  Google Scholar 

  4. Boucksom S. Divisorial Zariski decompositions on compact complex manifolds. Ann Sci Éc Norm Super (4), 2004, 37: 45–76

    Article  MathSciNet  Google Scholar 

  5. Boucksom S, Demailly J-P, Păun M, et al. The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension. J Algebraic Geom, 2013, 22: 201–248

    Article  MathSciNet  Google Scholar 

  6. Campana F, Flenner H. A characterization of ample vector bundles on a curve. Math Ann, 1990, 287: 571–575

    Article  MathSciNet  Google Scholar 

  7. Demailly J-P. Regularization of closed positive currents and intersection theory. J Algebraic Geom, 1992, 1: 361–409

    MathSciNet  MATH  Google Scholar 

  8. Demailly J-P. Complex Analytic and Differential Geometry. Http://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf, 2012

  9. Demailly J-P. Hermitian-Yang-Mills approach to the conjecture of Griffiths on the positivity of ample vector bundles. arXiv:2002.02677, 2020

  10. Demailly J-P, Peternell T, Schneider M. Compact complex manifolds with numerically effective tangent bundles. J Algebraic Geom, 1994, 3: 295–345

    MathSciNet  MATH  Google Scholar 

  11. Demailly J-P, Skoda H. Relations entre les notions de positivités de P. A. Griffiths et de S. Nakano pour les fibrés vectoriels. In: Séminaire Pierre Lelong-Henri Skoda (Analyse). Lecture Notes in Mathematics, vol. 822. Berlin-Heidelberg: Springer, 1980, 304–309

    Chapter  Google Scholar 

  12. Diverio S. Segre forms and Kobayashi-Lübke inequality. Math Z, 2016, 283: 1033–1047

    Article  MathSciNet  Google Scholar 

  13. Fu J X, Xiao J. Relations between the Kähler cone and the balanced cone of a Kähler manifold. Adv Math, 2014, 263: 230–252

    Article  MathSciNet  Google Scholar 

  14. Fulton W. Ample vector bundles, Chern classes, and numerical criteria. Invent Math, 1976, 32: 171–178

    Article  MathSciNet  Google Scholar 

  15. Fulton W. Intersection Theory, 2nd ed. Results in Mathematics and Related Areas, 3rd Series. A Series of Modern Surveys in Mathematics, vol. 2. Berlin: Springer-Verlag, 1998

    Book  Google Scholar 

  16. Fulton W, Lazarsfeld R. Positive polynomials for ample vector bundles. Ann of Math (2), 1983, 118: 35–60

    Article  MathSciNet  Google Scholar 

  17. Gieseker D. p-ample bundles and their Chern classes. Nagoya Math J, 1971, 43: 91–116

    Article  MathSciNet  Google Scholar 

  18. Griffiths P A. Hermitian differential geometry, Chern classes, and positive vector bundles. In: Global Analysis. Tokyo: University of Tokyo Press, 1969, 185–251

    Google Scholar 

  19. Guler D. Chern forms of positive vector bundles. PhD Thesis. Columbus: The Ohio State University, 2006

    Google Scholar 

  20. Guler D. On Segre forms of positive vector bundles. Canad Math Bull, 2012, 55: 108–113

    Article  MathSciNet  Google Scholar 

  21. Hartshorne R. Ample vector bundles. Publ Math Inst Hautes Études Sci, 1966, 29: 64–94

    Article  Google Scholar 

  22. Kleiman S L. Ample vector bundles on algebraic surfaces. Proc Amer Math Soc, 1969, 21: 673–676

    Article  MathSciNet  Google Scholar 

  23. Lazarsfeld R. Positivity in Algebraic Geometry II. Positivity for Vector Bundles, and Multiplier Ideals. Results in Mathematics and Related Areas, 3rd Series. A Series of Modern Surveys in Mathematics, vol. 49. Berlin: SpringerVerlag, 2004

    Google Scholar 

  24. Li P. Nonnegative Hermitian vector bundles and Chern numbers. Math Ann, 2021, 380: 21–41

    Article  MathSciNet  Google Scholar 

  25. Liu K F, Sun X F, Yang X K. Positivity and vanishing theorems for ample vector bundles. J Algebraic Geom, 2013, 22: 303–331

    Article  MathSciNet  Google Scholar 

  26. Mourougane C, Takayama S. Hodge metrics and positivity of direct images. J Reine Angew Math, 2007, 606: 167–178

    MathSciNet  MATH  Google Scholar 

  27. Nakayama N. Zariski-Decomposition and Abundance. MSJ Memoirs, vol. 14. Tokyo: Math Soc Japan, 2004

    MATH  Google Scholar 

  28. Naumann P. An approach to Griffiths conjecture. arXiv:1710.10034, 2017

  29. Nyström D W. Duality between the pseudoeffective and the movable cone on a projective manifold. J Amer Math Soc, 2019, 32: 675–689

    Article  MathSciNet  Google Scholar 

  30. Pingali V P. Representability of Chern-Weil forms. Math Z, 2018, 288: 629–641

    Article  MathSciNet  Google Scholar 

  31. Ross J, Toma M. Hodge-Riemann bilinear relations for Schur classes of ample vector bundles. arXiv:1905.13636, 2019

  32. Siu Y T. Analyticity of sets associated to Lelong numbers and the extension of closed positive currents. Invent Math, 1974, 27: 53–156

    Article  MathSciNet  Google Scholar 

  33. Toma M. A note on the cone of mobile curves. C R Acad Sci Ser I Math, 2010, 348: 71–73

    MathSciNet  MATH  Google Scholar 

  34. Umemura H. Some results in the theory of vector bundles. Nagoya Math J, 1973, 52: 97–128

    Article  MathSciNet  Google Scholar 

  35. Usui S, Tango H. On numerical positivity of ample vector bundles with additional condition. J Math Kyoto Univ, 1977, 17: 151–164

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work was supported by Tsinghua University Initiative Scientific Research Program (Grant No. 2019Z07L02016) and National Natural Science Foundation of China (Grant No. 11901336). The author thanks Ping Li for his constructive comments, and thanks the referees for careful reading and helpful comments.

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  1. Department of Mathematical Sciences and Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China

    Jian Xiao

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  1. Jian Xiao
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Xiao, J. On the positivity of high-degree Schur classes of an ample vector bundle. Sci. China Math. 65, 51–62 (2022). https://doi.org/10.1007/s11425-020-1868-7

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