Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 

 

The $T^1$-lifting theorem in positive characteristic


Author: Stefan Schröer
Journal: J. Algebraic Geom. 12 (2003), 699-714
DOI: https://doi.org/10.1090/S1056-3911-03-00330-8
Published electronically: July 2, 2003
MathSciNet review: 1993761
Full-text PDF

Abstract | References | Additional Information

Abstract: Replacing symmetric powers by divided powers and working over Witt vectors instead of ground fields, I generalize Kawamata's $T^1$-lifting theorem to characteristic $p>0$. Combined with the work of Deligne-Illusie on degeneration of the Hodge-de Rham spectral sequences, this gives unobstructedness for certain Calabi-Yau varieties with free crystalline cohomology modules.


References [Enhancements On Off] (What's this?)

  • 1. P. Berthelot, A. Ogus: Notes on crystalline cohomology. Princeton University Press, Princeton, 1978.
  • 2. F. Bogomolov: Hamiltonian Kählerian manifolds. Dokl. Akad. Nauk SSSR 243, 1101-1104 (1978); English transl., Soviet Math. Dokl. 19, 1462-1465 (1978).
  • 3. P. Deligne: Théorème de Lefschetz et critères de dégénérescence de suites spectrales. Publ. Math. Inst. Hautes Étud. Sci. 35, 107-126 (1968).
  • 4. P. Deligne, L. Illusie: Relèvements modulo $p\sp 2$ et décomposition du complexe de de Rham. Invent. Math. 89, 247-270 (1987).
  • 5. B. Fantechi, M. Manetti: Obstruction calculus for functors of Artin rings I. J. Algebra 202, 541-576 (1998).
  • 6. A. Grothendieck: Éléments de géométrie algébrique III: Étude cohomologique des faiscaux cohérents. Publ. Math., Inst. Hautes Étud. Sci. 17 (1963).
  • 7. A. Grothendieck: Éléments de géométrie algébrique IV: Étuede locale des schémas et de morphismes de schémas. Publ. Math., Inst. Hautes Étud. Sci. 20 (1964).
  • 8. A. Grothendieck et al.: Revêtements étales et groupe fondamental. Lect. Notes Math. 224, Springer, Berlin, 1971.
  • 9. A. Grothendieck: Catégories fibrées et descente. In SGA 1, pp. 145-194, Lect. Notes Math. 224. Springer, Berlin, 1971.
  • 10. N. Heerema: On ramified complete discrete valuation rings. Proc. Amer. Math. Soc. 10, 490-496 (1959).
  • 11. M. Hirokado: A non-liftable Calabi-Yau threefold in characteristic $3$. Tohoku Math. J. 51, 479-487 (1999).
  • 12. L. Illusie: Complexe cotangent et déformations I. Lect. Notes Math. 239, Springer, Berlin, 1971.
  • 13. L Illusie: Complexe de de Rham-Witt et cohomologie cristalline. Ann. Sci. Ecole Norm. Sup. 12, 501-661 (1979).
  • 14. Y. Kawamata: Unobstructed deformations. J. Algebraic Geom. 1, 183-190 (1992).
  • 15. Y. Kawamata, Y. Namikawa: Logarithmic deformations of normal crossing varieties and smoothing of degenerate Calabi-Yau varieties. Invent. Math. 118, 395-409 (1994).
  • 16. Y. Kawamata: Unobstructed deformations II. J. Algebraic Geom. 4, 277-279 (1995).
  • 17. Y. Kawamata: Erratum on: ``Unobstructed deformations." J. Algebraic Geom. 6, 803-804 (1997).
  • 18. H. Matsumura: Commutative ring theory. Cambridge Studies in Advanced Mathematics 8. Cambridge University Press, Cambridge, 1989.
  • 19. J. Milnor: Singular points of complex hypersurfaces. Annals of Mathematics Studies 61. Princeton University Press, Princeton, 1969.
  • 20. Z. Ran: Deformations of manifolds with torsion or negative canonical bundle. J. Algebraic Geom. 1, 279-291 (1992).
  • 21. D. Rim: Formal deformation theory. In SGA 7, pp. 32-132, Lect. Notes Math. 288. Springer, Berlin, 1972.
  • 22. M. Schlessinger: Functors of Artin rings. Trans. Amer. Math. Soc. 130, 208-222 (1968).
  • 23. G. Tian: Smoothness of the universal deformation space of compact Calabi-Yau manifolds and its Petersson-Weil metric. In: S. Yau (ed.), Mathematical aspects of string theory, pp. 629-646. Adv. Ser. Math. Phys. 1. World Sci. Publishing, Singapore, 1987.
  • 24. A. Todorov: Applications of the Kähler-Einstein-Calabi-Yau metric to moduli of $K3$ surfaces. Invent. Math. 61, 251-265 (1980).


Additional Information

Stefan Schröer
Affiliation: Mathematische Fakultät, Ruhr-Universität, 44780 Bochum, Germany
Address at time of publication: Mathematishes Institut, Universitaet Koeln, Weyertal 86-90, 50931 Koeln, Germany
Email: s.schroeer@ruhr-uni-bochum.de

DOI: https://doi.org/10.1090/S1056-3911-03-00330-8
Received by editor(s): March 6, 2001
Received by editor(s) in revised form: September 1, 2001
Published electronically: July 2, 2003

Journal of Algebraic Geometry
The Journal of Algebraic Geometry
is sponsored by the Department of Mathematical Sciences
of Tsinghua University
and is distributed by the American Mathematical Society
for University Press, Inc.
Online ISSN 1534-7486; Print ISSN 1056-3911
© 2017 University Press, Inc.
Comments: jag-query@ams.org
AMS Website