Skip to main content Accessibility help


  • KIRTI JOSHI (a1)


In this article, I give a crystalline characterization of abelian varieties amongst the class of smooth projective varieties with trivial tangent bundles in characteristic $p>0$ . Using my characterization, I show that a smooth, projective, ordinary variety with trivial tangent bundle is an abelian variety if and only if its second crystalline cohomology is torsion-free. I also show that a conjecture of KeZheng Li about smooth projective varieties with trivial tangent bundles in characteristic $p>0$ is true for smooth projective surfaces. I give a new proof of a result by Li and prove a refinement of it. Based on my characterization of abelian varieties, I propose modifications of Li’s conjecture, which I expect to be true.



Hide All
[1] Berthelot, P. and Ogus, A., Notes on crystalline cohomology, Math. Notes 21 , Princeton University Press, Princeton, 1978.
[2] Birkenhake, C. and Lange, H., Complex Abelian Varieties, Springer, 2004.
[3] Bombieri, E. and Mumford, D., “ Enriques’ classification of surfaces in characteristic p, II ”, Complex Analysis and Algebraic Geometry, 2342. Iwanami Shoten, Tokyo, 1977.
[4] Chai, C.-L., Mumfords example of non-flat $\operatorname{Pic}^{\unicode[STIX]{x1D70F}}$ . Unpublished,∼chai/papers_pdf/mumford_ex.pdf.
[5] Chai, C.-L., Conrad, B. and Oort, F., “ Complex multiplication and lifting problems ”, Mathematical Surveys and Monographs 197 , American Math. Society, 2014.
[6] Deligne, P. and Illusie, L., Relévements modulo p 2 et decomposition du complexe de de Rham , Invent. Math. 89(2) (1987), 247270.
[7] Faltings, G. and Chai, C.-L., Degeneration of Abelian Varieties, Ergebnisse de mathematik und ihrer grenzgebiete 22 , Springer, 1980.
[8] Fogarty, J. and Mumford, D., Geometric Invariant Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer.
[9] Hartshorne, R., Algebraic Geometry, Graduate Texts in Mathematics 52 , Springer, New York-Heidelberg, 1977.
[10] Igusa, J. i., Some problems in abstract algebraic geometry , Proc. Natl Acad. Sci. USA 41 (1955), 964967.
[11] Illusie, L., Complexe de de Rham–Witt et cohomologie cristalline , Ann. Sci. Éc. Norm. Supér. 12 (1979), 501661.
[12] Joshi, K., Exotic torsion, Frobenius splitting and the slope spectral sequence , Canad. Math. Bull. 50(4) (2007), 567578.
[13] Lang, S., Complex Multiplication, first edition, Grundlehren der mathematischen wissenschaften 255 , Springer, New York, 1983.
[14] Lange, H., Hyperelliptic varieties , Tohoku Math. J. 53 (2001), 491510.
[15] Li, KeZheng, Actions of group schemes , Compos. Math. 80 (1991), 5574.
[16] Li, KeZheng, Differential operators and automorphism schemes , Sci. China Math. 53(9) (2010), 23632380.
[17] Liedtke, C., A note on non-reduced Picard schemes , J. Pure Appl. Algebra 213 (2009), 737741.
[18] Mehta, V. B. and Srinivas, V., Varieties in positive characteristic with trivial tangent bundle , Compos. Math. 64 (1987), 191212.
[19] Ogus, A., Supersingular K3-crystals , Asterisque 64 (1979), 386.
[20] Yu, C.-F., The isomorphism classes of abelian varieties of CM-type , J. Pure Appl. Algebra 187 (2003), 305319.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Nagoya Mathematical Journal
  • ISSN: 0027-7630
  • EISSN: 2152-6842
  • URL: /core/journals/nagoya-mathematical-journal
Please enter your name
Please enter a valid email address
Who would you like to send this to? *

MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed