Abstract
We study curvature currents of singular Hermitian metrics on holomorphic vector bundles. It is known that curvature currents of singular Hermitian metrics on vector bundles are generally not defined with measure coefficients. In this paper, we give some sufficient conditions that curvature currents can be defined with measure coefficients. Moreover, we investigate Chern forms associated with singular Hermitian metrics.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Berman, R.J.: From Monge-Amp\(\grave{e}\)re equations to envelopes and geodesic rays in the zero temperature limit. Math. Z 1–30 (2018)
Berman, R. J., Demailly, J.-P.: Regularity of plurisubharmonic upper envelops in big cohomology classes. Perspectives in analysis, geometry, and topology. Birkhäuser/Springer, New York, pp. 39–66 (2012)
Berndtsson, B., Păun, M.: Bergman kernels and the pseudoeffectivity of relative canonical bundles. Duke Math. J. 145(2), 341–378 (2008)
Chu, J., Tosatti, V., Weinkove, B.: \(C^{1,1}\) regularity for degenerate complex \(\text{ Monge-Amp }\grave{e}\text{ re }\) equations and geodesics rays. Commun. Partial Differ. Equ. 43(2), 292–312 (2018)
Demailly, J.-P.: Analytic Methods in Algebraic Geometry, Surveys of Modern Mathematics, vol. 1. Higher Education Press, International Press, Beijing, Somerville (2012)
Demailly, J.-P., Peternell, T., Schneider, M.: Compact complex manifolds with numerically effective tangent bundles. J. Algebraic Geom. 3, 295–345 (1994)
Demailly, J.-P., Peternell, T., Schneider, M.: Pseudo-effective line bundles on compact Kähler manifolds. Int. J. Math. 12, 689–741 (2001)
Hacon, C., Popa, M., Schnell, C.: Algebraic fiber spaces over Abelian varieties : around a recent theorem by Cao and Paun, arXiv:1611.08768
Inayama, T.: \(L^2\) estimates and vanishing theorems for holomorphic vector bundles equipped with singular Hermitian metrics, arXiv:1802.10516
Lärkäng, R., Raufi, H., Ruppenthal, J., Sera, M.: Chern forms of singular metrics on vector bundles. Adv. Math. 326, 465–489 (2018)
Păun, M., Takayama, S.: Positivity of twisted relative pluricanonical bundles and their direct images. J. Algebraic Geom. 27, 211–272 (2018)
Raufi, H.: Singular hermitian metrics on holomorphic vector bundles. Ark. Mat. 53(2), 359–382 (2015)
Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton, NJ (1970)
Skoda, H.: Sous-ensembles analytiques d’ordre fini ou infini dans \(\mathbb{C}^n\). Bull. Soc. Math. Fr. 100, 353–408 (1972)
Yang, X.: Big vector bundles and complex manifolds with semi-positive tangent bundles. Math. Ann. 267(1), 251–282 (2017)
Acknowledgements
The author would like to thank Prof. Shigeharu Takayama and Prof. Bo Berndtsson for inspiring and valuable comments. He is also grateful to anonymous referees for their suggestions to improve the manuscript. This work is supported by the Program for Leading Graduate Schools, MEXT, Japan. This work is also supported by JSPS KAKENHI Grant Number 18J22119.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Inayama, T. Curvature Currents and Chern Forms of Singular Hermitian Metrics on Holomorphic Vector Bundles. J Geom Anal 30, 910–935 (2020). https://doi.org/10.1007/s12220-019-00164-9
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12220-019-00164-9