Classical Serre–Tate theory describes deformations of ordinary abelian varieties. It implies that every such variety has a canonical lift to characteristic zero and equips the base of its universal deformation with a Frobenius lifting and canonical multiplicative coordinates. A variant of this theory has been obtained for ordinary K3 surfaces by Nygaard and Ogus.
In this paper, we construct canonical liftings modulo of varieties with trivial canonical class which are ordinary in the weak sense that the Frobenius acts bijectively on the top cohomology of the structure sheaf. Consequently, we obtain a Frobenius lifting on the moduli space of such varieties. The quite explicit construction uses Frobenius splittings and a relative version of Witt vectors of length two. If the variety has unobstructed deformations and bijective first higher Hasse–Witt operation, the Frobenius lifting gives rise to canonical coordinates. One of the key features of our liftings is that the crystalline Frobenius preserves the Hodge filtration.
We also extend Nygaard’s approach from K3 surfaces to higher dimensions, and show that no non-trivial families of such varieties exist over simply connected bases with no global one-forms.
Funding statement: The first author was supported by NCN SONATA grant number 2017/26/D/ST1/00913. The second author’s work was supported by Zsolt Patakfalvi’s Swiss National Science Foundation Grant No. 200021/169639. This work was partially supported by the grant 346300 for IMPAN from the Simons Foundation and the matching 2015–2019 Polish MNiSW fund.
A Finite height
In , Yobuko defines the notion of a quasi-F-splitting, more general than an F-splitting, and proves that a smooth quasi-F-split variety can be lifted modulo , generalizing the argument for smooth F-split varieties [25, Section 8.5]. He also shows that for Calabi–Yau varieties, being quasi-F-split is equivalent to the height of the associated Artin–Mazur formal group being finite. In this section, we give a somewhat different point of view on Yobuko’s construction. Based on this, we extend our construction of a canonical lifting associated to an F-splitting to quasi-F-split varieties, thus showing that there is a preferred lifting mod . In particular, we show that the smoothness assumption used by Yobuko is not necessary. Some of our results were observed earlier by Adrian Langer (unpublished).
It would be interesting to extend this construction to families and to generalize the Serre–Tate theory discussed in this paper to varieties with trivial canonical class of finite height in the spirit of .
A.1 The canonical lifting
Yobuko has defined a quasi-F-splitting of level m on an -scheme X as an additive map satisfying and which is F-linear in the sense that
We have a short exact sequence
Suppose that is a quasi-F-splitting of level m on an -scheme X. Then the image of under V is an ideal in .
Suppose that satisfies , and let . Then
Suppose that is a quasi-F-splitting of level m on an -scheme X. Then the quotient
defines a lifting of X over , fitting inside a pushout diagram of exact sequences
Since σ is surjective, the left square is a pushout. To prove that is flat, we need to check that the composition
equals multiplication by p. Let be the image of
Then the image of in is , which is the image under σ of
This in turn has image in , which maps to under π. ∎
We call the lifting constructed in Corollary A.1.2 the canonical lifting associated to the quasi-F-splitting σ.
Composing the map with the Teichmüller lift yields a multiplicative section of . Consequently, the assertions of Lemma 5.2.2 hold for as well.
A.2 The sheaf and Witt vectors mod p
Suppose that X is a smooth scheme over a perfect field k of characteristic p. Denote by the Cartier operator. The sheaf consists of all local sections ω such that are closed, while the subsheaf of consists of all local sections ω such that . The map
is -linear, and there is a short exact sequence of -modules
In this situation, Yobuko defined a natural extension of -modules (denoted )
and showed that -linear splittings of this sequence correspond to quasi-F-splittings as defined previously. The exact sequence is defined by the pullback diagram
In particular, if X is F-split, the bottom row is, and hence so is . Yobuko observes that can also be defined as a pushout
It follows that quasi-F-splittings are precisely -linear splittings of .
We shall now elucidate the sheaf and its -algebra structure. To this end, let X be an -scheme. Following [20, Section 3], we denote by
the mod-p reduction of the ring of Witt vectors of length m over . Recall from [20, Section 3] that the p-th power of the Teichmüller map
is additive modulo p, and therefore induces an -algebra structure
on . In addition, the following triangle commutes:
Consequently, ρ is injective if X is reduced. We denote its cokernel by , so that there is a short exact sequence of -modules
implies that the multiplication in is highly degenerate.
The existence of ρ can be also seen as follows (cf. [47, Section 1.1]): the kernel of has a natural divided power structure, and hence so does the kernel I of
It follows that for every , and hence the absolute Frobenius of factors naturally through .
The following diagram is a pushout square:
The commutativity of the diagram follows from the formula
Since the vertical arrows are surjective, it remains to check that the induced map
is an isomorphism. This follows from
Suppose that X is a smooth scheme over k. There are natural isomorphisms
fitting inside a commutative diagram
We can therefore rephrase the definition of a quasi-F-splitting as follows.
Let X be an -scheme. A quasi-F-splitting of level m on X is an -linear splitting of the map .
B Families with vanishing Kodaira–Spencer
Let k be a perfect field of characteristic and let S be a smooth k-scheme. Let be a smooth morphism. Applying to the short exact sequence on X
yields an exact sequence on S
The Kodaira–Spencer map of is the map obtained as the composition of the adjunction map and .
Theorem B.0.1 (compare [45, Lemma 3.5]).
Suppose and . Then there exists a canonical cartesian diagram
The assumptions and combined with the exactness of (B.2) show that the adjunction map factors uniquely through a map . By another adjunction, we obtain a map which splits (B.1). We check that v defines a 1-foliation (see [12, Chapter I, Section 1, p. 104]), i.e. that its image is closed under the Lie bracket and p-th iterates. The respective obstructions [12, Lemma 1.4, p. 105] are maps
by adjunction correspond to maps
which both vanish because the target is zero. Define Y to be the quotient by this 1-foliation. ∎
As the example and shows, the assumption that is necessary. Note that in this example still descends along Zariski-locally on S. To produce examples which do not descend even locally, one can take a Brauer–Severi variety whose corresponding class in is not divisible by p.
Of course, if , then the adjunction map can locally be lifted to a map (since is locally free). However, for the above proof to work, we need the lifting u to be compatible with the restricted Lie algebra structures.
We would like to thank Bhargav Bhatt, Jędrzej Garnek, Adrian Langer, Daniel Litt, Arthur Ogus, Lenny Taelman, and Jakub Witaszek for helpful discussions. We thank the referees for many helpful comments regarding the manuscript. Part of this work was conducted during the first author’s stay at the MPIM and the Hausdorff Center for Mathematics in Bonn.
 P. Achinger, J. Witaszek and M. Zdanowicz, Global Frobenius liftability II: Surfaces and Fano threefolds, preprint (2017), https://arxiv.org/abs/2102.02788. 10.2422/2036-2145.202005_003Search in Google Scholar
 P. Achinger, J. Witaszek and M. Zdanowicz, Global Frobenius liftability I, J. Eur. Math. Soc. (JEMS) 23 (2021), no. 8, 2601–2648. 10.4171/JEMS/1063Search in Google Scholar
 M. Artin and B. Mazur, Formal groups arising from algebraic varieties, Ann. Sci. Éc. Norm. Supér. (4) 10 (1977), no. 1, 87–131. 10.24033/asens.1322Search in Google Scholar
 P. Berthelot and A. Ogus, Notes on crystalline cohomology, Princeton University, Princeton 1978. Search in Google Scholar
 S. Bloch and K. Kato, p-adic étale cohomology, Publ. Math. Inst. Hautes Études Sci. 63 (1986), 107–152. 10.1007/978-1-4757-9284-3_2Search in Google Scholar
 F. A. Bogomolov, Hamiltonian Kählerian manifolds, Dokl. Akad. Nauk SSSR 243 (1978), no. 5, 1101–1104. Search in Google Scholar
 J. Borger and L. Gurney, Canonical lifts of families of elliptic curves, Nagoya Math. J. 233 (2019), 193–213. 10.1017/nmj.2017.34Search in Google Scholar
 E. Costa, Effective computations of Hasse–Weil zeta functions, ProQuest LLC, Ann Arbor 2015; Ph.D. thesis, New York University, New York 2015. Search in Google Scholar
 S. A. Cynk and D. van Straten, Small resolutions and non-liftable Calabi–Yau threefolds, Manuscripta Math. 130 (2009), no. 2, 233–249. 10.1007/s00229-009-0293-0Search in Google Scholar
 P. Deligne, Cristaux ordinaires et coordonnées canoniques, Algebraic surfaces (Orsay 1976–78), Lecture Notes in Math. 868, Springer, Berlin (1981), 80–137; With the collaboration of L. Illusie, With an appendix by Nicholas M. Katz. 10.1007/BFb0090647Search in Google Scholar
 P. Deligne and L. Illusie, Relèvements modulo et décomposition du complexe de de Rham, Invent. Math. 89 (1987), no. 2, 247–270. 10.1007/BF01389078Search in Google Scholar
 T. Ekedahl, Canonical models of surfaces of general type in positive characteristic, Publ. Math. Inst. Hautes Études Sci. 67 (1988), 97–144. 10.1007/BF02699128Search in Google Scholar
 T. Ekedahl, J. M. E. Hyland and N. I. Shepherd-Barron, Moduli and periods of simply connected Enriques surfaces, preprint (2012), https://arxiv.org/abs/1210.0342. Search in Google Scholar
 T. Ekedahl and N. I. Shepherd-Barron, Tangent lifting of deformations in mixed characteristic, J. Algebra 291 (2005), no. 1, 108–128. 10.1016/j.jalgebra.2005.05.023Search in Google Scholar
 H. Esnault and A. Shiho, Convergent isocrystals on simply connected varieties, Ann. Inst. Fourier (Grenoble) 68 (2018), no. 5, 2109–2148. 10.5802/aif.3204Search in Google Scholar
 L. R. A. Finotti, Lifting the j-invariant: Questions of Mazur and Tate, J. Number Theory 130 (2010), no. 3, 620–638. 10.1016/j.jnt.2009.10.009Search in Google Scholar
 M. Gross and B. Siebert, Mirror symmetry via logarithmic degeneration data. I, J. Differential Geom. 72 (2006), no. 2, 169–338. 10.4310/jdg/1143593211Search in Google Scholar
 A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III, Publ. Math. Inst. Hautes Études Sci. 28 (1966), 5–255. 10.1007/BF02684343Search in Google Scholar
 R. Hartshorne, Algebraic geometry, Grad. Texts in Math. 52, Springer, New York 1977. 10.1007/978-1-4757-3849-0Search in Google Scholar
 L. Hesselholt and I. Madsen, On the K-theory of local fields, Ann. of Math. (2) 158 (2003), no. 1, 1–113. 10.4007/annals.2003.158.1Search in Google Scholar
 M. Hirokado, A non-liftable Calabi–Yau threefold in characteristic 3, Tohoku Math. J. (2) 51 (1999), no. 4, 479–487. 10.2748/tmj/1178224716Search in Google Scholar
 D. Huybrechts, Fourier–Mukai transforms in algebraic geometry, Oxford Math. Monogr., The Clarendon Press, Oxford 2006. 10.1093/acprof:oso/9780199296866.001.0001Search in Google Scholar
 L. Illusie (Ed.), Cohomologie l-adique et fonctions L, Lecture Notes in Math. 589, Springer, Berlin 1977. Search in Google Scholar
 L. Illusie, Complexe de de Rham–Witt et cohomologie cristalline, Ann. Sci. Éc. Norm. Supér. (4) 12 (1979), no. 4, 501–661. 10.24033/asens.1374Search in Google Scholar
 L. Illusie, Frobenius et dégénérescence de Hodge, Introduction à la théorie de Hodge, Panor. Synthèses 3, Société Mathématique de France, Paris (1996), 113–168. Search in Google Scholar
 L. Illusie, Grothendieck’s existence theorem in formal geometry, Fundamental algebraic geometry, Math. Surveys Monogr. 123, American Mathematical Society, Providence (2005), 179–233. Search in Google Scholar
 L. Illusie and M. Raynaud, Les suites spectrales associées au complexe de de Rham–Witt, Inst. Hautes Études Sci. Publ. Math. 57 (1983), 73–212. 10.1007/BF02698774Search in Google Scholar
 N. Katz, Travaux de Dwork, Séminaire Bourbaki, 24ème année (1971/1972). Exp. No. 409, Lecture Notes in Math. 317, Springer, Berlin (1973), 167–200. 10.1007/BFb0069282Search in Google Scholar
 N. Katz, Serre–Tate local moduli, Algebraic surfaces (Orsay 1976–78), Lecture Notes in Math. 868, Springer, Berlin (1981), 138–202. 10.1007/BFb0090648Search in Google Scholar
 N. M. Katz, Algebraic solutions of differential equations (p-curvature and the Hodge filtration), Invent. Math. 18 (1972), 1–118. 10.1007/BF01389714Search in Google Scholar
 N. M. Katz, Slope filtration of F-crystals, Journées de géométrie algébrique de Rennes. Vol. I (Rennes 1978), Astérisque 63, Société Mathématique de France, Paris (1979), 113–163. Search in Google Scholar
 L. Katzarkov, M. Kontsevich and T. Pantev, Hodge theoretic aspects of mirror symmetry, From Hodge theory to integrability and TQFT tt*-geometry, Proc. Sympos. Pure Math. 78, American Mathematical Society, Providence (2008), 87–174. 10.1090/pspum/078/2483750Search in Google Scholar
 Y. Kawamata, Unobstructed deformations. A remark on a paper of Z. Ran, J. Algebraic Geom. 1 (1992), no. 2, 183–190. Search in Google Scholar
 H. Lange and U. Stuhler, Vektorbündel auf Kurven und Darstellungen der algebraischen Fundamentalgruppe, Math. Z. 156 (1977), no. 1, 73–83. 10.1007/BF01215129Search in Google Scholar
 C. Liedtke, Arithmetic moduli and lifting of Enriques surfaces, J. reine angew. Math. 706 (2015), 35–65. 10.1515/crelle-2013-0068Search in Google Scholar
 H. Matsumura, Commutative ring theory, 2nd ed., Cambridge Stud. Adv. Math. 8, Cambridge University, Cambridge 1989.Search in Google Scholar
 B. Mazur, Frobenius and the Hodge filtration (estimates), Ann. of Math. (2) 98 (1973), 58–95. 10.2307/1970906Search in Google Scholar
 V. B. Mehta and A. Ramanathan, Frobenius splitting and cohomology vanishing for Schubert varieties, Ann. of Math. (2) 122 (1985), no. 1, 27–40. 10.2307/1971368Search in Google Scholar
 V. B. Mehta and V. Srinivas, Varieties in positive characteristic with trivial tangent bundle, Compos. Math. 64 (1987), no. 2, 191–212. Search in Google Scholar
 S. Mochizuki, A theory of ordinary p-adic curves, Publ. Res. Inst. Math. Sci. 32 (1996), no. 6, 957–1152. 10.2977/prims/1195145686Search in Google Scholar
 L. Moret-Bailly, Pinceaux de variétés abéliennes, Astérisque 129, Société Mathématique de France, Paris 1985. Search in Google Scholar
 N. Nitsure, Construction of Hilbert and Quot schemes, Math. Surveys Monogr. 123, American Mathematical Society, Providence (2005), 105–137. Search in Google Scholar
 N. Nygaard and A. Ogus, Tate’s conjecture for surfaces of finite height, Ann. of Math. (2) 122 (1985), no. 3, 461–507. 10.2307/1971327Search in Google Scholar
 N. O. Nygaard, The Tate conjecture for ordinary surfaces over finite fields, Invent. Math. 74 (1983), no. 2, 213–237. 10.1007/BF01394314Search in Google Scholar
 A. Ogus, F-crystals and Griffiths transversality, Proceedings of the International Symposium on Algebraic Geometry (Kyoto 1977), Kinokuniya Book, Tokyo (1978), 15–44. Search in Google Scholar
 A. Ogus, Griffiths transversality in crystalline cohomology, Ann. of Math. (2) 108 (1978), no. 2, 395–419. 10.2307/1971182Search in Google Scholar
 A. Ogus and V. Vologodsky, Nonabelian Hodge theory in characteristic p, Publ. Math. Inst. Hautes Études Sci. 106 (2007), 1–138. 10.1007/s10240-007-0010-zSearch in Google Scholar
 Z. Ran, Deformations of manifolds with torsion or negative canonical bundle, J. Algebraic Geom. 1 (1992), no. 2, 279–291. Search in Google Scholar
 S. Schröer, The -lifting theorem in positive characteristic, J. Algebraic Geom. 12 (2003), no. 4, 699–714. 10.1090/S1056-3911-03-00330-8Search in Google Scholar
 S. Schröer, Some Calabi–Yau threefolds with obstructed deformations over the Witt vectors, Compos. Math. 140 (2004), no. 6, 1579–1592. 10.1112/S0010437X04000545Search in Google Scholar
 S. Schröer, The Deligne–Illusie theorem and exceptional Enriques surfaces, Eur. J. Math. 7 (2021), no. 2, 489–525. 10.1007/s40879-021-00451-2Search in Google Scholar
 E. Sernesi, Deformations of algebraic schemes, Grundlehren Math. Wiss. 334, Springer, Berlin 2006. Search in Google Scholar
 N. I. Shepherd-Barron, Weyl group covers for Brieskorn’s resolutions in all characteristics and the integral cohomology of , preprint (2017), https://arxiv.org/abs/1711.10439. 10.1307/mmj/1593741747Search in Google Scholar
 G. Tian, Smoothness of the universal deformation space of compact Calabi–Yau manifolds and its Petersson–Weil metric, Mathematical aspects of string theory (San Diego 1986), Adv. Ser. Math. Phys. 1, World Scientific, Singapore (1987), 629–646. 10.1142/9789812798411_0029Search in Google Scholar
 A. N. Todorov, The Weil–Petersson geometry of the moduli space of (Calabi–Yau) manifolds. I, Comm. Math. Phys. 126 (1989), no. 2, 325–346. 10.1007/BF02125128Search in Google Scholar
 G. van der Geer and T. Katsura, On the height of Calabi–Yau varieties in positive characteristic, Doc. Math. 8 (2003), 97–113. Search in Google Scholar
 M. Ward, Arithmetic properties of the derived category for Calabi–Yau varieties, ProQuest LLC, Ann Arbor 2014; Ph.D. thesis, University of Washington, Seattle 2014. Search in Google Scholar
 C. A. Weibel, An introduction to homological algebra, Cambridge Stud. Adv. Math. 38, Cambridge University, Cambridge 1994. 10.1017/CBO9781139644136Search in Google Scholar
 F. Yobuko, Quasi-Frobenius splitting and lifting of Calabi–Yau varieties in characteristic p, Math. Z. 292 (2019), no. 1–2, 307–316. 10.1007/s00209-018-2198-7Search in Google Scholar
 M. Zdanowicz, Liftability of singularities and their Frobenius morphism modulo , Int. Math. Res. Not. IMRN 14 (2018), 4513–4577. 10.1093/imrn/rnw297Search in Google Scholar
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