Abstract
We study conditions of Hörmander’s L2-estimate and the Ohsawa-Takegoshi extension theorem. Introducing a twisted version of the Hörmander-type condition, we show a converse of Hörmander L2-estimate under some regularity assumptions on an n-dimensional domain. This result is a partial generalization of the 1-dimensional result obtained by Berndtsson (1998). We also de_ne new positivity notions for vector bundles with singular Hermitian metrics by using these conditions. We investigate these positivity notions and compare them with classical positivity notions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Berndtsson B. Prekopa’s theorem and Kiselman’s minimum principle for plurisubharmonic functions. Math Ann, 1998, 312: 785–792
Berndtsson B, Păaun M. Bergman kernels and the pseudoeffectivity of relative canonical bundles. Duke Math J, 2008, 145: 341–378
Demailly J-P. Complex analytic and differential geometry. http://www-fourier.ujf-grenoble.fr/demailly/manuscripts/agbook.pdf, 2012
Demailly J-P. Analytic Methods in Algebraic Geometry. Surveys of Modern Mathematics, vol. 1. Somerville: International Press, 2012
Demailly J-P, Skoda H. Relations entre les notions de positivité de P. A. Griffiths et de S. Nakano. In: Séminaire Pierre Lelong-Henri Skoda (Analyse) Années. Lecture Notes in Mathematics, vol. 822. Berlin: Springer-Verlag, 1980, 304–309
Deng F, Ning J, Wang Z. Characterizations of plurisubharmonic functions. ArXiv:1910.06518, 2019
Deng F, Ning J, Wang Z, et al. Positivity of holomorphic vector bundles in terms of -Lp-conditions of ∂̄. ArXiv: 2001.01762, 2020
Deng F, Wang Z, Zhang L, et al. New characterizations of plurisubharmonic functions and positivity of direct image sheaves. ArXiv:1809.10371, 2018
Deng F, Wang Z, Zhang L, et al. Linear invariants of complex manifolds and their plurisubharmonic variations. ArXiv:1901.08920, 2019
Deng F, Zhang H, Zhou X. Positivity of direct images of positively curved volume forms. Math Z, 2014, 278: 347–362
Hacon C, Popa M, Schnell C. Algebraic fiber spaces over Abelian varieties: Around a recent theorem by Cao and Paun. Contemp Math, 2018, 712: 143–195
Hosono G. Approximations and examples of singular Hermitian metrics on vector bundles. Ark Mat, 2017, 55: 131–153
Păun M, Takayama S. Positivity of twisted relative pluricanonical bundles and their direct images. J Algebraic Geom, 2018, 27: 211–272
Raufi H. Singular hermitian metrics on holomorphic vector bundles. Ark Mat, 2015, 53: 359–382
Acknowledgements
This work was supported by the Program for Leading Graduate Schools, the Ministry of Education, Culture, Sports, Science and Technology, Japan, and Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research (Grant No. 18J22119). The authors thank their supervisor Professor Shigeharu Takayama for inspiring and helpful comments.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hosono, G., Inayama, T. A converse of Hörmander’s L2-estimate and new positivity notions for vector bundles. Sci. China Math. 64, 1745–1756 (2021). https://doi.org/10.1007/s11425-019-1654-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-019-1654-9