Skip to main content
SpringerLink
Log in

Logarithmic Kodaira dimension and zeros of holomorphic log-one-forms

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

In this paper, we prove that the zero-locus of any global holomorphic log-one-form on a projective log-smooth pair (X; D) of log-general type must be non-empty.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abramovich, D., Karu, K.: Weak semistable reduction in characteristic 0. Invent. Math. 139(2), 241–273 (2000)

    Article  MathSciNet  Google Scholar 

  2. Birkar, C., Cascini, P., Hacon, C.D., McKernan, J.: Existence of minimal models for varieties of log general type. J. Am. Math. Soc. 23(2), 405–468 (2010)

    Article  MathSciNet  Google Scholar 

  3. Fujino, O.: On quasi-Albanese maps (2014). https://www.math.kyoto-u.ac.jp/~fujino/quasi-albanese2.pdf (preprint)

  4. Hartshorne, R.: Algebraic Geometry, Graduate Texts in Mathematics, No. 52. Springer, New York (1977)

  5. Iitaka, S.: Logarithmic forms of algebraic varieties. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 23(3), 525–544 (1976)

    MathSciNet  MATH  Google Scholar 

  6. Kawamata, Y.: Characterization of abelian varieties. Compos. Math. 43(2), 253–276 (1981)

    MathSciNet  MATH  Google Scholar 

  7. Kovács, S., Patakfalvi, Z.: Projectivity of the moduli space of stable log-varieties and subadditivity of log-kodaira dimension. J. Am. Math. Soc. 30(4), 959–1021 (2017)

    Article  MathSciNet  Google Scholar 

  8. Luo, T., Zhang, Q.: Holomorphic forms on threefolds. In: Recent progress in arithmetic and algebraic geometry, Contemp. Math., vol. 386, pp. 87–94. Amer. Math. Soc., Providence (2005)

  9. Mumford, D.: Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5. Tata Institute of Fundamental Research, Bombay, Hindustan Book Agency, New Delhi (2008) (With appendices by C. P. Ramanujam and Yuri Manin, Corrected reprint of the second (1974) edition)

  10. Popa, M.: Kodaira–Saito vanishing and applications. Enseign. Math. 62(1–2), 49–89 (2016)

    Article  MathSciNet  Google Scholar 

  11. Popa, M., Schnell, C.: Generic vanishing theory via mixed Hodge modules. Forum Math. Sigma 1(60), e1 (2013)

    Article  MathSciNet  Google Scholar 

  12. Popa, M., Schnell, C.: Kodaira dimension and zeros of holomorphic one-forms. Ann. Math. 179(3), 1109–1120 (2014)

    Article  MathSciNet  Google Scholar 

  13. Popa, M., Schnell, C.: Viehweg’s hyperbolicity conjecture for families with maximal variation. Invent. Math. 208(3), 677–713 (2017)

    Article  MathSciNet  Google Scholar 

  14. Saito, M.: Modules de Hodge polarisables. Publ. Res. Inst. Math. Sci. 24(6), 849–995 (1989). 1988

    Article  MathSciNet  Google Scholar 

  15. Saito, M.: Mixed Hodge modules. Publ. Res. Inst. Math. Sci. 26(2), 221–333 (1990)

    Article  MathSciNet  Google Scholar 

  16. Sabbah, C., Schnell, C.: The MHM project (2016). http://www.cmls.polytechnique.fr/perso/sabbah/MHMProject/mhm.html (preprint)

  17. Taji, B.: The isotriviality of smooth families of canonically polarized manifolds over a special quasi-projective base. Compos. Math. 152(7), 1421–1434 (2016)

    Article  MathSciNet  Google Scholar 

  18. Viehweg, E., Zuo, K.: On the isotriviality of families of projective manifolds over curves. J. Algebraic Geom. 10(4), 781–799 (2001)

    MathSciNet  MATH  Google Scholar 

  19. Wei, C.: Fibration of log-general type space over quasi-abelian varieties (2016). arXiv:1609.03089

  20. Wei, C.: Logarithmic comparison theorems in mixed Hodge modules. Mich. Math. J. 69(1), 201–223 (2020)

    Article  Google Scholar 

  21. Wei, C., Wu, L.: Hyperbolicity for log smooth families with maximal variation. arXiv:1811.07466 (preprint)

  22. Wei, C., Wu, L.: Isotriviality of smooth families of varieties of general type. arXiv:2001.08360 (preprint)

Download references

Acknowledgements

The author would like to express his gratitude to his advisor Christopher Hacon for suggesting this topic and useful discussions. The author thanks Honglu Fan, Kalle Karu, Mihnea Popa, Christian Schnell, Lei Wu and Ziwen Zhu for answering his questions, and especially Schnell for suggesting a better choice of Hodge module which is used in the paper that essentially simplifies the proof of the main theorem. During the preparation of this paper, the author was partially supported by Hacon’s Grant DMS-1300750, DMS-1265285 and a grant from the Simons Foundation, Award Number 256202.

Author information

Authors and Affiliations

  1. Stony Brook University, Stony Brook, USA

    Chuanhao Wei

Authors
  1. Chuanhao Wei
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chuanhao Wei.

Additional information

Communicated by Vasudevan Srinivas.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wei, C. Logarithmic Kodaira dimension and zeros of holomorphic log-one-forms. Math. Ann. 378, 485–512 (2020). https://doi.org/10.1007/s00208-020-02031-3

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-020-02031-3

Search

Navigation