Skip to main content
SpringerLink
Log in

A Kawamata–Viehweg type formulation of the logarithmic Akizuki–Nakano vanishing theorem

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

In this article, we present a Kawamata–Viehweg type formulation of the (logarithmic) Akizuki–Nakano Vanishing Theorem. We give two proofs: one by reduction to an older theorem of Steenbrink via Kawamata’s covering lemma, and another by mod p reduction using results of Deligne–Illusie and Hara. We also include two applications.

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. Akizuki, Y., Nakano, S.: Note on Kodaira–Spencer’s proof of Lefschetz’s theorem. Proc. Jpn. Acad. Ser. A 30, 266–272 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  2. Arapura, D.: Kodaira-Saito vanishing via Higgs bundles in positive characteristic. J. Reine Angew. Math. (To appear)

  3. Arapura, D.: Local cohomology of sheaves of differential forms and Hodge theory. J. Reine Angew. Math. 409, 172–179 (1990)

    MathSciNet  MATH  Google Scholar 

  4. Arapura, D.: Frobenius amplitude and strong vanishing theorems for vector bundles. With an appendix by Dennis S. Keeler. Duke Math. J. 121(2), 231–267 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Deligne, P.: Théorie de Hodge. II. Inst. Hautes Études Sci. Publ. Math. 40, 5–57 (1971)

    Article  MATH  Google Scholar 

  6. Deligne, P., Illusie, L.: Relevetments modulo \(p^2\) et decomposition du complexe de de Rham. Invent. Math. 89, 247–280 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  7. Esnault, H., Viehweg, E.: Logarithmic de Rham complexes and vanishing theorems. Invent. Math. 86(1), 161–194 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  8. Esnault, H., Viehweg, E.: Lectures on Vanishing Theorems, DMV Seminar, vol. 20. Birkhäuser Verlag, Basel (1992)

    Book  MATH  Google Scholar 

  9. Flenner, H.: Extendability of differential forms on nonisolated singularities. Invent. Math. 94(2), 317–326 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  10. Guillen, F., Navarro, A.V., Pascual, G.P., Puerta, F.: Hyperrésolutions cubiques et descente cohomologique. Papers from the Seminar on Hodge–Deligne Theory Held in Barcelona, Lecture Notes in Mathematics, vol. 1335, p. 1988. Springer, Berlin (1982)

    Google Scholar 

  11. Hacon, C., Kovács, S.: Holomorphic one-forms on varieties of general type. Ann. Sci. École Norm. Sup. (4) 38(4), 599–607 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hara, N.: A characterization of rational singularities in terms of injectivity of Frobenius maps. Am. J. Math. 120, 981–996 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  13. Huang, C., Liu, K., Wan X., Yang, X.: Logarithmic vanishing theorems on compact Kähler manifolds I. arXiv:1611.07671 (2016)

  14. Kawamata, Y.: On a generalization of Kodaira–Ramanujam’s vanishing theorem. Math. Ann. 261, 43–46 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kawamata, Y., Namikawa, Y.: Logarithmic deformations of normal crossing varieties and smoothing of degenerate Calabi–Yau varieties. Invent. Math. 118(3), 395–409 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kawamata, Y., Matsuda, K., Matsuki, K.: Introduction to the minimal model problem. Adv. Stud. Pure Math. 10, 283–360 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kodaira, K.: On a differential-geometric method in the theory of analytic stacks. Proc. Natl. Acad. Sci. USA 39, 1268–1273 (1953)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kollár, J.: Vanishing theorems for cohomology groups, Algebraic Geometry Bowdoin 1985. Proceedings of Symposium Pure Math., vol. 46, pp. 233–243 (1987)

  19. Kovács, S.: On the number of singular fibers in a family of surfaces of general type. J. Reine Angew. Math. 487, 171–177 (1997)

    MathSciNet  MATH  Google Scholar 

  20. Kovács, S.: Logarithmic vanishing theorems and Arakelov–Parshin boundedness for singular varieties. Compos. Math. 131(3), 291–317 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kovács, S.: Steenbrink vanishing extended. Bull. Braz. Math. Soc. (N.S.) 45(4), 753–765 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  22. Lazarsfeld, R.: Positivity in Algebraic Geometry I. Springer, Berlin (2004)

    Book  MATH  Google Scholar 

  23. Matsuki, K.: Introduction to the Mori Program. Universitext, Springer, Berlin (2002)

    Book  MATH  Google Scholar 

  24. Matsuki, K., Olsson, M.: Kawamata–Viehweg vanishing as Kodaira vanishing for stacks. Math. Res. Lett. 12, 207–217 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  25. Mori, S., Mukai, S.: Classification of Fano 3-folds with \(B_2 \ge 2\). Manusc. Math. 36(2), 147–162 (1981)

    Article  MATH  Google Scholar 

  26. Norimatsu, Y.: Kodaira vanishing theorem and Chern classes for \(\partial \)- manifolds. Proc. Jpn. Acad. Ser. A Math. Sci. 54(4), 107–108 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  27. Popa, M., Schnell, C.: Kodaira dimension and zeros of holomorphic one-forms. Ann. Math. (2) 179(3), 1109–1120 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  29. Sernesi, E.: Defomormations of Algebraic Schemes, Grundlehren der Mathematischen Wissenschaften, vol. 334. Springer, Berlin (2006)

    Google Scholar 

  30. Steenbrink, J., Vanishing theorems on singular spaces. Differential systems and singularities (Luminy, 1983) Asterisque, No. 130 (1985)

  31. Viehweg, E.: Vanishing theorems. J. Reine Angew. Math. 335, 1–8 (1982)

    MathSciNet  MATH  Google Scholar 

  32. 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 

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

  34. Wei, C.: Logarithmic Kodaira dimension and zeros of holomorphic log-one-forms. arXiv:1711.05854v1 [math], November 15 (2017)

  35. Zhang, Q.: Global holomorphic one-forms on projective manifolds with ample canonical bundles. J. Algebraic Geom. 6(4), 777–787 (1997)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We thank Professors O. Fujino, C. Hacon, S. Helmke, Y. Kawamata, Y. Namikawa, A. Moriwaki, S. Mori, S. Mukai, and M. Nori for invaluable comments, suggestions and support. We also thank the members of our seminar, P. Coupek, H. Li, and H. Wang for listening to the talks about the subject.

Author information

Authors and Affiliations

  1. Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN, 47907, USA

    Donu Arapura, Kenji Matsuki, Deepam Patel & Jarosław Włodarczyk

Authors
  1. Donu Arapura
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kenji Matsuki
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Deepam Patel
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jarosław Włodarczyk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Deepam Patel.

Additional information

Publisher's Note

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

D.A. was partially supported by a grant from the Simons foundation. K.M. would like to thank RIMS at Kyoto for their warm hospitality and generous support. D.P. would like to acknowledge support from the National Science Foundation award DMS-1502296. J.W. would like to thank the Binational (U.S. - Israel) Science Fund BSF 2014365 for their support.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Arapura, D., Matsuki, K., Patel, D. et al. A Kawamata–Viehweg type formulation of the logarithmic Akizuki–Nakano vanishing theorem. Math. Z. 303, 83 (2023). https://doi.org/10.1007/s00209-023-03225-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00209-023-03225-6

Search

Navigation