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Singular hermitian metrics on holomorphic vector bundles

Abstract

We introduce and study the notion of singular hermitian metrics on holomorphic vector bundles, following Berndtsson and Păun. We define what it means for such a metric to be positively and negatively curved in the sense of Griffiths and investigate the assumptions needed in order to locally define the curvature Θh as a matrix of currents. We then proceed to show that such metrics can be regularised in such a way that the corresponding curvature tensors converge weakly to Θh. Finally we define what it means for h to be strictly negatively curved in the sense of Nakano and show that it is possible to regularise such metrics with a sequence of smooth, strictly Nakano negative metrics.

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References

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

    MathSciNet  Article  Google Scholar 

  2. Berndtsson, B. and Păun, M., Bergman kernels and the pseudoeffectivity of relative canonical bundles, Duke Math. J. 145 (2008), 341–378.

    MathSciNet  Article  MATH  Google Scholar 

  3. de Cataldo, M. A. A., Singular Hermitian metrics on vector bundles, J. Reine Angew. Math. 502 (1998), 93–122.

    MathSciNet  Google Scholar 

  4. Demailly, J.-P., Estimations L 2 pour l’opérateur d’un fibré vectoriel holomorphe semi-positif au-dessus d’une variété kählérienne complète, Ann. Sci. Éc. Norm. Supér. 15 (1982), 457–511.

    MathSciNet  MATH  Google Scholar 

  5. Demailly, J.-P., Singular Hermitian metrics on positive line bundles, in Complex Algebraic Varieties (Bayreuth, 1990), Lecture Notes in Math. 1507, pp. 87–104, Springer, Berlin, 1992.

    Chapter  Google Scholar 

  6. Nadel, A. M., Multiplier ideal sheaves and Kähler–Einstein metrics of positive scalar curvature, Ann. of Math. 132 (1990), 549–596.

    MathSciNet  Article  MATH  Google Scholar 

  7. Raufi, H., The Nakano vanishing theorem and a vanishing theorem of Demailly–Nadel type for vector bundles, Preprint, Göteborg, 2012. arXiv:1212.4417.

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Correspondence to Hossein Raufi.

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Raufi, H. Singular hermitian metrics on holomorphic vector bundles. Ark Mat 53, 359–382 (2015). https://doi.org/10.1007/s11512-015-0212-4

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  • DOI: https://doi.org/10.1007/s11512-015-0212-4

Keywords

  • Vector Bundle
  • Line Bundle
  • Curvature Tensor
  • Holomorphic Section
  • Plurisubharmonic Function