Abstract
Let \(f:X\rightarrow Y\) be a smooth fibration between two complex manifolds X and Y, and let L be a pseudo-effective line bundle on X. We obtain a sufficient condition for \(R^{q}f_{*}(K_{X/Y}\otimes L)\) to be reflexive and hence derive a Kollár-type vanishing theorem.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ambro, F.: An injectivity theorem. Compos. Math. 150, 999–1023 (2014)
Berndtsson, B.: The openness conjecture and complex Brunn-Minkowski inequalities. In: Complex Geometry and Dynamics. Abel Symposia, vol. 10, pp. 29–44. Springer, Cham (2015)
Demailly, J.-P.: Estimations \(L^{2}\) pour l’opérateur \(\bar{\partial }\) d’un fibré vectoriel holomorphe semi-positif au-dessus d’une variété kählérienne complète. Ann. Sci. École Norm. Sup. 4(15), 457–511 (1982)
Demailly, J.-P.: Analytic methods in algebraic geometry. In: Surveys of Modern Mathematics, vol. 1. International Press, Somerville, MA; Higher Education Press, Beijing (2012). ISBN: 978-1-57146-234-3
Demailly, J.-P., Peternell, T., Schneider, M.: Pseudo-effective line bundles on compact Kähler manifolds. Internat J. Math. 6, 689–741 (2001)
Esnault, H., Viehweg, E.: Lectures on vanishing theorems. DMV Seminar, 20. Birkhäuser, Basel (1992). ISBN: 3-7643-2822-3
de Fernex, T., Ein, L., Mustaţă, M.: Vanishing theorems and singularities in birational geometry. http://homepages.math.uic.edu/~ein/DFEM.pdf
Fujino, O.: A transcendental approach to Kollár’s injectivity theorem. Osaka J. Math. 49, 833–852 (2012)
Gongyo, Y., Matsumura, S.: Versions of injectivity and extension theorems. Ann. Sci. Éc. Norm. Supér. 4(50), 479–502 (2017)
Hartshorne, R.: Stable reflexive sheaves. Math. Ann. 254, 121–176 (1980)
Kollár, J.: Higher direct images of dualizing sheaves. I. Ann. Math. 2(123), 11–42 (1986)
Matsumura, S.: A vanishing theorem of Kollár–Ohsawa type. Math. Ann. 366, 1451–1465 (2016)
Matsumura, S.: An injectivity theorem with multiplier ideal sheaves of singular metrics with transcendental singularities. J. Algebra. Geom. 27, 305–337 (2018)
Mourougane, C., Takayama, S.: Hodge metrics and the curvature of higher direct images. Ann. Sci. Éc. Norm. Sup. 4(41), 905–924 (2008)
Mumford, D.: Lectures on Curves on an Algebraic Surface. With a section by G. M. Bergman. Annals of Mathematics Studies, vol. 59. Princeton University Press, Princeton, NJ (1966)
Ohsawa, T.: Vanishing theorems on complete Kähler manifolds. Publ. Res. Inst. Math. Sci. 20, 21–38 (1984)
Ohsawa, T.: Analysis of several complex variables. Translated from the Japanese by Shu Gilbert Nakamura. Translations of Mathematical Monographs, 211. Iwanami Series in Modern Mathematics. American Mathematical Society, Providence, RI (2002). ISBN: 0-8218-2098-2
Siu, Y.-T.: A vanishing theorem for semipositive line bundles over non-Kähler manifolds. J. Differ. Geom. 19, 431–452 (1984)
Takegoshi, K.: Higher direct images of canonical sheaves tensorized with semi-positive vector bundles by proper Kähler morphisms. Math. Ann. 303, 389–416 (1995)
Acknowledgements
The author sincerely thanks his supervisor Professor Jixiang Fu for discussions. Thanks also go to Shin-ichi Matsumura, who kindly provided some comments about the references of this paper. Finally, he is very grateful to the referee for many useful suggestions on how to improve the paper.
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
Wu, J. A Kollár-type vanishing theorem. Math. Z. 295, 331–340 (2020). https://doi.org/10.1007/s00209-019-02406-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-019-02406-6