Abstract
We study the Fujita-type conjecture proposed by Popa and Schnell. We obtain an effective bound on the global generation of direct images of pluri-adjoint line bundles on the regular locus. We also obtain an effective bound on the generic global generation for a Kawamata log canonical \(\mathbb {Q}\)-pair. We use analytic methods such as \(L^2\) estimates, \(L^2\) extensions and injective theorems of cohomology groups.
Similar content being viewed by others
References
Angehrn, U., Siu, Y.-T.: Effective freeness and point separation for adjoint bundles. Invent. Math. 122(2), 291–308 (1995)
Berndtsson, B., Păun, M.: Bergman kernels and the pseudoeffectivity of relative canonical bundles. Duke Math. J. 145(2), 341–378 (2008)
Cao, J.-Y.: Ohsawa–Takegoshi extension theorem for compact Kähler manifolds and applications. Springer INdAM Ser. 21, 19–38 (2017)
Cao, J.-Y., Demailly, J.-P., Matsumura, S.: A general extension theorem for cohomology classes on non reduced analytic subspaces. Sci. China Math. 60(6), 949–962 (2017)
Demailly, J.-P.: Analytic methods in algebraic geometry. Surv. Modern Math. 1, 231 (2012)
Deng, Y.: Applications of the Ohsawa–Takegoshi extension theorem to direct image problems. arXiv:1703.07279v2
Dutta, Y.: On the effective freeness of the direct images of pluricanonical bundles. arXiv:1701.08830v3
Dutta, Y., Murayama, T.: Effective generation and twisted weak positivity of direct images. arXiv:1712.08723v1
Hacon, C., Popa, M., Schnell, C.: Algebraic fiber spaces over abelian varieties: around a recent theorem by Cao and Paun. arXiv:1611.08768v2
Kollár, J., Mori, S.: Birational Geometry of Algebraic Varieties. With the Collaboration of C. H. Clemens and A. Corti. Translated from the 1998 Japanese original. Cambridge Tracts in Mathematics, p. viii+254. Cambridge University Press, Cambridge (1998)
Lazarsfeld, R.: Positivity in Algebraic Geometry. II. Positivity for Vector Bundles, and Multiplier Ideals Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge A Series of Modern Surveys in Mathematics (Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics), vol. 49, p. 385. Springer, Berlin (2004)
Popa, M., Schnell, C.: On direct images of pluricanonical bundles. Algebra Number Theory. 8(9), 2273–2295 (2014)
Păun, M., Takayama, S.: Positivity of twisted relative pluricanonical bundles and their direct images. arXiv:1409.5504v1(to appear in J. Algebraic Geom.)
Acknowledgements
The author would like to thank his supervisor Prof. Shigeharu Takayama for helpful comments and enormous support. He wishes to thank Genki Hosono for several useful discussions. He also wishes to thank Yajnaseni Dutta for letting him know their results in [8] and sending their preprint. He is greatly indebted to the referee for many kind advices. This work is supported by the Program for Leading Graduate Schools, MEXT, Japan. This work is also supported by JSPS KAKENHI Grant Number 17J04457.
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
Iwai, M. On the global generation of direct images of pluri-adjoint line bundles. Math. Z. 294, 201–208 (2020). https://doi.org/10.1007/s00209-019-02266-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-019-02266-0