Skip to main content

When is the Albanese morphism an algebraic fiber space in positive characteristic?

An Erratum to this article was published on 23 April 2022

This article has been updated


In this paper, we study the Albanese morphisms in positive characteristic. We prove that the Albanese morphism of a variety with nef anti-canonical divisor is an algebraic fiber space, under the assumption that the general fiber is F-pure. Furthermore, we consider a notion of F-splitting for morphisms, and investigate it in the case of Albanese morphisms. We show that an F-split variety has F-split Albanese morphism, and that the F-split Albanese morphism is an algebraic fiber space. As an application, we provide a new characterization of abelian varieties.

This is a preview of subscription content, access via your institution.

Change history


  1. Atiyah, M.F.: On the Krull–Schmidt theorem with application to sheaves. Bull. Soc. Math. Fr. 84, 307–317 (1956)

    Article  MathSciNet  Google Scholar 

  2. Atiyah, M.F.: Vector bundles over an elliptic curve. Proc. Lond. Math. Soc. 7, 414–452 (1957)

    Article  MathSciNet  Google Scholar 

  3. Bombieri, E., Mumford, D.: Enriques’ Classification of Surfaces in Char.\(p\), II. Complex Analysis and Algebraic Geometry (Dedicated to K. Kodaira) Part I, pp. 23–42. Iwanami Shoten, Tokyo (1977)

    Google Scholar 

  4. Cao, J.: Albanese maps of projective manifolds with nef anticanonical bundles. arXiv:1612.05921, to appear in Ann. Sci. École Norm. Sup, (2016)

  5. Conrad, B.: Grothendieck Duality and Base Change, Volume 1750 of Lecture Notes in Mathematics. Springer, Berlin (2000)

    Google Scholar 

  6. de Jong, A.J.: Smoothness, semi-stability and alterations. Publ. Math. Inst. Hautes Études Sci. 83(1), 51–93 (1996)

    Article  MathSciNet  Google Scholar 

  7. Ejiri, S.: Positivity of anti-canonical divisors and \(f\)-purity of fibers. arXiv preprintarXiv:1604.02022 (2016)

  8. Ejiri, S.: Weak positivity theorem and Frobenius stable canonical rings of geometric generic fibers. J. Algebr. Geom. 26, 691–734 (2017)

    Article  MathSciNet  Google Scholar 

  9. Ejiri, S., Sannai, A.: A characterization of ordinary abelian varieties by Frobenius push-forward of the structure sheaf \(II\) (2017). arXiv:1702.04209, to appear in Int. Math. Res. Not

  10. Fantechi, B., Göttsche, L., Illusie, L., Kleiman, S.L., Nitsure, N., Vistoli, A.: Fundamental Algebraic Geometry: Grothendieck’s FGA Explained, vol. 123. American Mathematical Society, Providence (2005)

    MATH  Google Scholar 

  11. Gongyo, Y., Okawa, S., Sannai, A., Takagi, S.: Characterization of varieties of Fano type via singularities of Cox rings. J. Algebr. Geom. 24(1), 159–182 (2015)

    Article  MathSciNet  Google Scholar 

  12. Gongyo, Y., Takagi, S.: Surfaces of globally \(F\)-regular and \(F\)-split type. Math. Ann. 364(3), 841–855 (2016)

    Article  MathSciNet  Google Scholar 

  13. Hacon, C.D., Patakfalvi, Z.: Generic vanishing in characteristic \(p>0\) and the characterization of ordinary abelian varieties. Am. J. Math. 138(4), 963–998 (2016)

    Article  MathSciNet  Google Scholar 

  14. Hacon, C.D., Patakfalvi, Z.: On the characterization of abelian varieties in characteristic \( p> 0\) (2016). arXiv:1602.01791

  15. Hacon, C.D., Patakfalvi, Z., Zhang, L.: Birational characterization of abelian varieties and ordinary abelian varieties in characteristic \(p>0\) (2017). arXiv:1703.06631

  16. Hartshorne, R.: Residues and Duality, Volume 20 of Lecture Notes in Mathematics. Springer, Berlin (1966)

    Google Scholar 

  17. Hashimoto, M.: \(F\)-pure homomorphisms, strong \(F\)-regularity, and \(F\)-injectivity. Commun. Algebra 38(12), 4569–4596 (2010)

    Article  MathSciNet  Google Scholar 

  18. Iitaka, S.: Algebraic Geometry—An Introduction to Birational Geometry of Algebraic Varieties, Volume 76 of Graduate Texts in Mathematics. Springer, New York (1982)

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  20. Lange, H., Stuhler, U.: Vektorbündel auf kurven und darstellungen fundamentalgruppe. Math Z. 156, 73–83 (1977)

    Article  MathSciNet  Google Scholar 

  21. Lu, S., Tu, Y., Zhang, Q., Zheng, Q.: On semistability of albanese maps. Manuscr. Math. 131(3), 531–535 (2010)

    Article  MathSciNet  Google Scholar 

  22. Mehta, V.B., Nori, M.V.: Semistable sheaves on homogeneous spaces and abelian varieties. In: Proceedings of the Indian Academy of Sciences-Mathematical Sciences, vol. 93, pp. 1–12. Springer (1984)

  23. Mehta, V.B., Ramanathan, A.: Frobenius splitting and cohomology vanishing for schubert varieties. Ann. Math. 122(1), 27–40 (1985)

    Article  MathSciNet  Google Scholar 

  24. Mehta, V.B., Srinivas, V.: Varieties in positive characteristic with trivial tangent bundle. Compos. Math. 64(2), 191–212 (1987)

    MathSciNet  MATH  Google Scholar 

  25. Mumford, D.: Abelian Varieties. Oxford University Press, Oxford (1974)

    MATH  Google Scholar 

  26. Oda, T.: Vector bundles on an elliptic curve. Nagoya Math. J. 43, 41–72 (1971)

    Article  MathSciNet  Google Scholar 

  27. Okawa, S.: Surfaces of globally \(F\)-regular type are of Fano type. Tohoku Math. J. 69(1), 35–42 (2017)

    Article  MathSciNet  Google Scholar 

  28. Sannai, A., Tanaka, H.: A characterization of ordinary abelian varieties by the Frobenius push-forward of the structure sheaf. Math. Ann. 366(3–4), 1067–1087 (2016)

    Article  MathSciNet  Google Scholar 

  29. Schwede, K.: A canonical linear system associated to adjoint divisors in characteristic \(p>0\). J. Reine Angew. Math. 696, 69–87 (2014)

    MathSciNet  MATH  Google Scholar 

  30. Schwede, K., Smith, K.E.: Globally \(F\)-regular and log Fano varieties. Adv. Math. 224(3), 863–894 (2010)

    Article  MathSciNet  Google Scholar 

  31. Viehweg, E.: Weak Positivity and the Additivity of the Kodaira Dimension for Certain Fiber Spaces, pp. 329–353. Kinokuniya, North-Holland (1983)

    MATH  Google Scholar 

  32. Wang, Y.: On the characterization of abelian varieties for log pairs in zero and positive characteristic (2016). arXiv:1610.05630

  33. Zhang, Q.: On projective varieties with nef anticanonical divisors. Math. Ann. 332(3), 697–703 (2005)

    Article  MathSciNet  Google Scholar 

Download references


The author wishes to express his gratitude to his supervisor Professor Shunsuke Takagi for suggesting problems, valuable comments and helpful advice. He is deeply grateful to Professors Zsolt Patakfalvi and Yoshinori Gongyo for fruitful discussions and valuable comments. He would like to thank Professors Osamu Fujino, Nobuo Hara and Doctor Yuan Wang for stimulating discussions, questions and comments. He also would like to thank the reviewer for a careful reading and helpful suggestions. Part of this work was carried out during his visit to Princeton University with support from The University of Tokyo/Princeton University Strategic Partnership Teaching and Research Collaboration Grant, and from the Program for Leading Graduate Schools, MEXT, Japan. He was also supported by JSPS KAKENHI Grant Number 15J09117.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Sho Ejiri.

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ejiri, S. When is the Albanese morphism an algebraic fiber space in positive characteristic?. manuscripta math. 160, 239–264 (2019).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

Mathematics Subject Classification