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
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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.
Berthelot;Ogus1978 P. Berthelot, A. Ogus: Notes on crystalline cohomology. Princeton University Press, Princeton, 1978.
Bogomolov1978 F. Bogomolov: Hamiltonian Kählerian manifolds. Dokl. Akad. Nauk SSSR 243, 1101–1104 (1978); English transl., Soviet Math. Dokl. 19, 1462–1465 (1978).
Deligne1968 P. Deligne: Théorème de Lefschetz et critères de dégénérescence de suites spectrales. Publ. Math. Inst. Hautes Étud. Sci. 35, 107–126 (1968).
Deligne;Illusie1987 P. Deligne, L. Illusie: Relèvements modulo $p^ 2$ et décomposition du complexe de de Rham. Invent. Math. 89, 247–270 (1987).
Fantechi;Manetti1998 B. Fantechi, M. Manetti: Obstruction calculus for functors of Artin rings I. J. Algebra 202, 541–576 (1998).
EGAIIIb A. Grothendieck: Éléments de géométrie algébrique III: Étude cohomologique des faiscaux cohérents. Publ. Math., Inst. Hautes Étud. Sci. 17 (1963).
EGAIVa A. Grothendieck: Éléments de géométrie algébrique IV: Étuede locale des schémas et de morphismes de schémas. Publ. Math., Inst. Hautes Étud. Sci. 20 (1964).
SGA1 A. Grothendieck et al.: Revêtements étales et groupe fondamental. Lect. Notes Math. 224, Springer, Berlin, 1971.
Grothendieck1971 A. Grothendieck: Catégories fibrées et descente. In SGA 1, pp. 145–194, Lect. Notes Math. 224. Springer, Berlin, 1971.
Heerema1959 N. Heerema: On ramified complete discrete valuation rings. Proc. Amer. Math. Soc. 10, 490–496 (1959).
Hirokado1999 M. Hirokado: A non-liftable Calabi–Yau threefold in characteristic $3$. Tohoku Math. J. 51, 479–487 (1999).
Illusie1971 L. Illusie: Complexe cotangent et déformations I. Lect. Notes Math. 239, Springer, Berlin, 1971.
Illusie1979 L Illusie: Complexe de de Rham–Witt et cohomologie cristalline. Ann. Sci. Ecole Norm. Sup. 12, 501–661 (1979).
Kawamata1992 Y. Kawamata: Unobstructed deformations. J. Algebraic Geom. 1, 183–190 (1992).
Kawamata;Namikawa1994 Y. Kawamata, Y. Namikawa: Logarithmic deformations of normal crossing varieties and smoothing of degenerate Calabi-Yau varieties. Invent. Math. 118, 395–409 (1994).
Kawamata1995 Y. Kawamata: Unobstructed deformations II. J. Algebraic Geom. 4, 277–279 (1995).
Kawamata1997 Y. Kawamata: Erratum on: “Unobstructed deformations." J. Algebraic Geom. 6, 803–804 (1997).
Matsumura1989 H. Matsumura: Commutative ring theory. Cambridge Studies in Advanced Mathematics 8. Cambridge University Press, Cambridge, 1989.
Milnor1968 J. Milnor: Singular points of complex hypersurfaces. Annals of Mathematics Studies 61. Princeton University Press, Princeton, 1969.
Ran1992 Z. Ran: Deformations of manifolds with torsion or negative canonical bundle. J. Algebraic Geom. 1, 279–291 (1992).
Rim1969 D. Rim: Formal deformation theory. In SGA 7, pp. 32–132, Lect. Notes Math. 288. Springer, Berlin, 1972.
Schlessinger1968 M. Schlessinger: Functors of Artin rings. Trans. Amer. Math. Soc. 130, 208–222 (1968).
Tian1987 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.
Todorov1980 A. Todorov: Applications of the Kähler–Einstein–Calabi–Yau metric to moduli of $K3$ surfaces. Invent. Math. 61, 251–265 (1980).
Berthelot;Ogus1978 P. Berthelot, A. Ogus: Notes on crystalline cohomology. Princeton University Press, Princeton, 1978.
Bogomolov1978 F. Bogomolov: Hamiltonian Kählerian manifolds. Dokl. Akad. Nauk SSSR 243, 1101–1104 (1978); English transl., Soviet Math. Dokl. 19, 1462–1465 (1978).
Deligne1968 P. Deligne: Théorème de Lefschetz et critères de dégénérescence de suites spectrales. Publ. Math. Inst. Hautes Étud. Sci. 35, 107–126 (1968).
Deligne;Illusie1987 P. Deligne, L. Illusie: Relèvements modulo $p^ 2$ et décomposition du complexe de de Rham. Invent. Math. 89, 247–270 (1987).
Fantechi;Manetti1998 B. Fantechi, M. Manetti: Obstruction calculus for functors of Artin rings I. J. Algebra 202, 541–576 (1998).
EGAIIIb A. Grothendieck: Éléments de géométrie algébrique III: Étude cohomologique des faiscaux cohérents. Publ. Math., Inst. Hautes Étud. Sci. 17 (1963).
EGAIVa A. Grothendieck: Éléments de géométrie algébrique IV: Étuede locale des schémas et de morphismes de schémas. Publ. Math., Inst. Hautes Étud. Sci. 20 (1964).
SGA1 A. Grothendieck et al.: Revêtements étales et groupe fondamental. Lect. Notes Math. 224, Springer, Berlin, 1971.
Grothendieck1971 A. Grothendieck: Catégories fibrées et descente. In SGA 1, pp. 145–194, Lect. Notes Math. 224. Springer, Berlin, 1971.
Heerema1959 N. Heerema: On ramified complete discrete valuation rings. Proc. Amer. Math. Soc. 10, 490–496 (1959).
Hirokado1999 M. Hirokado: A non-liftable Calabi–Yau threefold in characteristic $3$. Tohoku Math. J. 51, 479–487 (1999).
Illusie1971 L. Illusie: Complexe cotangent et déformations I. Lect. Notes Math. 239, Springer, Berlin, 1971.
Illusie1979 L Illusie: Complexe de de Rham–Witt et cohomologie cristalline. Ann. Sci. Ecole Norm. Sup. 12, 501–661 (1979).
Kawamata1992 Y. Kawamata: Unobstructed deformations. J. Algebraic Geom. 1, 183–190 (1992).
Kawamata;Namikawa1994 Y. Kawamata, Y. Namikawa: Logarithmic deformations of normal crossing varieties and smoothing of degenerate Calabi-Yau varieties. Invent. Math. 118, 395–409 (1994).
Kawamata1995 Y. Kawamata: Unobstructed deformations II. J. Algebraic Geom. 4, 277–279 (1995).
Kawamata1997 Y. Kawamata: Erratum on: “Unobstructed deformations." J. Algebraic Geom. 6, 803–804 (1997).
Matsumura1989 H. Matsumura: Commutative ring theory. Cambridge Studies in Advanced Mathematics 8. Cambridge University Press, Cambridge, 1989.
Milnor1968 J. Milnor: Singular points of complex hypersurfaces. Annals of Mathematics Studies 61. Princeton University Press, Princeton, 1969.
Ran1992 Z. Ran: Deformations of manifolds with torsion or negative canonical bundle. J. Algebraic Geom. 1, 279–291 (1992).
Rim1969 D. Rim: Formal deformation theory. In SGA 7, pp. 32–132, Lect. Notes Math. 288. Springer, Berlin, 1972.
Schlessinger1968 M. Schlessinger: Functors of Artin rings. Trans. Amer. Math. Soc. 130, 208–222 (1968).
Tian1987 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.
Todorov1980 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
MR Author ID:
630946
Email:
s.schroeer@ruhr-uni-bochum.de
Received by editor(s):
March 6, 2001
Received by editor(s) in revised form:
September 1, 2001
Published electronically:
July 2, 2003
