Abstract
Building on the results of Deligne and Illusie on liftings to truncated Witt vectors, we give a criterion for non-liftability that involves only the dimension of certain cohomology groups of vector bundles arising from the Frobenius pushforward of the de Rham complex. Using vector bundle methods, we apply this to show that exceptional Enriques surfaces, a class introduced by Ekedahl and Shepherd-Barron, do not lift to truncated Witt vectors, yet the base of the miniversal formal deformation over the Witt vectors is regular. Using the classification of Bombieri and Mumford, we also show that bielliptic surfaces arising from a quotient by a unipotent group scheme of order p do not lift to the ring of Witt vectors. These results hinge on some observations in homological algebra that relates splittings in derived categories to Yoneda extensions and certain diagram completions.
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References
Anantharaman, S.: Schémas en groupes, espaces homogènes et espaces algébriques sur une base de dimension 1. Mémoires de la Société Mathématique de France, vol. 33. Société Mathématique de France, Paris (1973)
Artin, M.: Algebraic construction of Brieskorn’s resolutions. J. Algebra 29, 330–348 (1974)
Barakat, M., Bremer, B.: Higher extension modules and the Yoneda product (2008). arXiv:0802.3179
Berthelot, P., Grothendieck, A., Illusie, L. (eds.): Théorie des Intersections et Théorème de Riemann–Roch (SGA 6). Lecture Notes in Mathematics, vol. 225. Springer, Berlin (1971)
Bogomolov, F.A.: Hamiltonian Kählerian manifolds. Dokl. Akad. Nauk SSSR 243(5), 1101–1104 (1978) (in Russian)
Bourbaki, N.: Algèbre Commutative. Chapitre 8–9. Masson, Paris (1983)
Bombieri, E., Mumford, D.: Enriques’ classification of surfaces in char \(p\). III. Invent. Math. 35, 197–232 (1976)
Bombieri, E., Mumford, D.: Enriques’ classification of surfaces in char \(p\), II. In: Baily, W., Shioda, T. (eds.) Complex Analysis and Algebraic Geometry, pp. 23–42. Cambridge University Press, London (1977)
Bosch, S., Lütkebohmert, W., Raynaud, M.: Néron Models. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 21. Springer, Berlin (1990)
Bourbaki, N.: Algèbre. Chapitres 1 à 3. Hermann, Paris (1970)
Brion, M., Kumar, S.: Frobenius Splitting Methods in Geometry and Representation Theory. Progress in Mathematics, vol. 231. Birkhäuser, Boston (2005)
Cossec, F.R., Dolgachev, I.V.: Enriques Surfaces. Vol. I. Progress in Mathematics, vol. 76. Birkhäuser, Boston (1989)
De Clercq, C., Florence, M., Lucchini Arteche, G.: Lifting vector bundles to Witt vector bundles (2018). arXiv:1807.04859
Deligne, P.: Relèvement des surfaces K3 en caractéristique nulle. In: Giraud, J., Illusie, L., Raynaud, M. (eds.), Surface Algébrique, pp. 58–79. Lecture Notes in Mathematics, vol. 868. Springer, Berlin (1981)
Deligne, P., Illusie, L.: Relèvements modulo \(p^2\) et décomposition du complexe de de Rham. Invent. Math. 89(2), 247–270 (1987)
Demazure, M., Grothendieck, A. (eds.): Schémas en groupes II (SGA 3 Tome 2). Lecture Notes in Mathematics, vol. 152. Springer, Berlin (1970)
Demazure, M., Gabriel, P.: Groupes Algébriques. Masson, Paris (1970)
Ekedahl, T., Hyland, J.M.E., Shepherd-Barron, N.I.: Moduli and periods of simply connected Enriques surfaces (2012). arXiv:1210.0342
Ekedahl, T., Shepherd-Barron, N.I.: Tangent lifting of deformations in mixed characteristic. J. Algebra 291(1), 108–128 (2005)
Ekedahl, T., Shepherd-Barron, N.I.: On exceptional Enriques surfaces. arXiv:math/0405510
Fanelli, A., Schröer, S.: Del Pezzo surfaces and Mori fiber spaces in positive characteristic. Trans. Amer. Math. Soc. 373(3), 1775–1843 (2020)
Gabriel, P., Zisman, M.: Calculus of Fractions and Homotopy Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 35. Springer, New York (1967)
Giraud, J.: Cohomologie Non Abélienne. Die Grundlehren der mathematischen Wissenschaften, vol. 179. Springer, Berlin (1971)
Grothendieck, A.: Éléments de géométrie algébrique II: Étude globale élémentaire de quelques classes de morphismes. Publ. Math. Inst. Hautes Étud. Sci. 8 (1961)
Grothendieck, A.: Les schémas de Picard: Propriétés générales. Séminaire Bourbaki, Exp. 236, 221–243 (1962)
Grothendieck, A.: Revêtements Étales et Groupe Fondamental (SGA 1). Lecture Notes in Mathematics, vol. 224. Springer, Berlin (1971)
Hartshorne, R.: Local Cohomology. Lecture Notes in Mathematics, vol. 41. Springer, Berlin (1967)
Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics, vol. 52. Springer, New York (1977)
Hartshorne, R.: Generalized divisors on Gorenstein schemes. K-Theory 8(3), 287–339 (1994)
Illusie, L.: Complexe de de Rham–Witt et cohomologie cristalline. Ann. Sci. Ecole Norm. Sup. 12(4), 501–661 (1979)
Kashiwara, M., Schapira, P.: Categories and Sheaves. Grundlehren der Mathematischen Wissenschaften, vol. 332. Springer, Berlin (2006)
Katz, N.M.: Nilpotent connections and the monodromy theorem: applications of a result of Turrittin. Inst. Hautes Études Sci. Publ. Math. 39, 175–232 (1970)
Kawamata, Y.: Unobstructed deformations. J. Algebraic Geom. 1(2), 183–190 (1992)
Kondō, S., Schröer, S.: Kummer surfaces associated with group schemes. Manuscripta. Math. https://doi.org/10.1007/s00229-020-01257-4
Laumon, G., Moret-Bailly, L.: Champs Algebriques. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 39. Springer, Berlin (2000)
Liedtke, C.: Arithmetic moduli and lifting of Enriques surfaces. J. Reine Angew. Math. 706, 35–65 (2015)
Liedtke, C., Satriano, M.: On the birational nature of lifting. Adv. Math. 254, 118–137 (2014)
Matsumura, H.: Commutative Ring Theory. Cambridge Studies in Advanced Mathematics, vol. 8. Cambridge University Press, Cambridge (1989)
McLean, K.R.: Commutative artinian principal ideal rings. Proc. London Math. Soc. 26, 249–272 (1973)
Mehta, V., Ramanathan, A.: Frobenius splitting and cohomology vanishing for Schubert varieties. Ann. Math. 122(1), 27–40 (1985)
Mitchell, B.: Theory of Categories. Pure and Applied Mathematics, vol. 27. Academic Press, New York (1965)
Olsson, M.: Algebraic Spaces and Stacks. American Mathematical Society Colloquium Publications, vol. 62. American Mathematical Society, Providence (2016)
Oort, F., Mumford, D.: Deformations and liftings of finite, commutative group schemes. Invent. Math. 5, 317–334 (1968)
Partsch, H.: Deformations of elliptic fiber bundles in positive characteristic. Nagoya Math. J. 211, 79–108 (2013)
Ran, Z.: Deformations of manifolds with torsion or negative canonical bundle. J. Algebraic Geom. 1(2), 279–291 (1992)
Salomonsson, P.: Equations for some very special Enriques surfaces in characteristic two (2003). arXiv:math/0309210
Schlessinger, M.: Functors of artin rings. Trans. Amer. Math. Soc. 130, 208–222 (1968)
Schröer, S.: The \(T^1\)-lifting theorem in positive characteristics. J. Algebraic Geom. 12(4), 699–714 (2003)
Schröer, S.: Kummer surfaces for the self-product of the cuspidal rational curve. J. Algebraic Geom. 16(2), 305–346 (2007)
Schröer, S.: Enriques surfaces with normal K3-like coverings (2017). arXiv:1703.03081. J. Math. Soc. Japan (to appear)
Shepherd-Barron, N.: Weyl group covers for Brieskorn’s resolutions in all characteristics and the integral cohomology of \(G/P\). Michigan Math. J. https://doi.org/10.1307/mmj/1593741747
Srinivas, V.: Decomposition of the de Rham complex. Proc. Indian Acad. Sci. Math. Sci. 100(2), 103–106 (1990)
Tian, G.: Smoothness of the universal deformation space of compact Calabi–Yau manifolds and its Petersson–Weil metric. In: Yau, S. (ed.) Mathematical Aspects of String Theory, pp. 629–646. World Scientific, Singapore (1987)
Todorov, A.N.: Applications of the Kähler–Einstein–Calabi–Yau metric to moduli of \(K3\) surfaces. Invent. Math. 61(3), 251–265 (1980)
Yobuko, F.: Quasi-frobenius-splitting and lifting of Calabi–Yau varieties in characteristic \(p\). Math. Z. 292(1–2), 307–316 (2019)
Yoneda, N.: On the homology theory of modules. J. Fac. Sci. Univ. Tokyo. Sect. I(7), 193–227 (1954)
Zariski, O., Samuel, P.: Commutative Algebra, vol. I. Van Nostrand, Princeton (1958)
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I wish to thank Luc Illusie for valuable discussions, and the referees for thorough reading and helpful comments.
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This research was conducted in the framework of the research training group GRK 2240: Algebro-geometric Methods in Algebra, Arithmetic and Topology, which is funded by the DFG.
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Schröer, S. The Deligne–Illusie Theorem and exceptional Enriques surfaces. European Journal of Mathematics 7, 489–525 (2021). https://doi.org/10.1007/s40879-021-00451-2
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DOI: https://doi.org/10.1007/s40879-021-00451-2