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The ordinary limit for varieties over $$\mathbb{Z}[x_1,\dots,x_r]$$. (English) Zbl 1067.14019
Yui, Noriko (ed.) et al., Calabi-Yau varieties and mirror symmetry. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3355-3/hbk). Fields Inst. Commun. 38, 273-305 (2003).
This is the second installment of a series of papers addressing certain aspects of arithmetic geometry of Calabi-Yau threefolds. This article is the sequel to a paper in the same volume [in: Calabi-Yau varieties and mirror symmetry, Fields Inst. Commun. 38, 255–271 (2003; Zbl 1063.14023)]. In an archimedean environment one works with complex geometry and complex functions and studies Calabi-Yau threefolds near the large complex structure limit. This paper proposes to look at Calabi-Yau threefolds in a non-archimedean environment, which is represented by projective systems of groups $$\{M_N\}_{N\in{\mathbb N}}$$ with $$M_N$$ a module over $${\mathbb Z}/n{\mathbb Z}$$, indexed by the positive integers with their divisibility relation.
This paper focuses on the question: What is the right notion of “ordinariness” for varieties over $${\mathbb Z}[x_1,\dots, x_r]?$$ This may be illustrated in terms of formal groups associated to these varieties. For dimension $$1$$ and $$2$$ Calabi-Yau varieties in finite characteristic, the ordinariness is characterized by the invertibility of the Hasse-Witt invariant. However, for Calabi-Yau threefolds, invertibility of the Hasse-Witt invariant is necessary but not sufficient for ordinariness. The technical core of the paper is the conjugate filtration and ordinariness. For families of smooth projective varieties over a localized polynomial ring $${\mathbb Z}[x_1,\dots, x_r][D^{-1}]$$, the conjugate filtration on de Rham cohomology $$\otimes {\mathbb Z}/N{\mathbb Z}$$ is studied in detail. As $$N\to\infty$$, this leads to the concept of the ordinary limit, which is a good candidate for the non-archimedean analogue of the large complex structure limit.
In the appendix, Artin-Mazur formal groups arising from Calabi-Yau varieties are discussed. The Hasse-Witt invariant of a certain Calabi-Yau variety (e.g., a complete intersection) can be computed explicitly via the logarithm for the Artin-Mazur formal group. One striking observation is that the holomorphic solution of the Picard-Fuchs differential equation near the large complex structure limit shows up in the ordinary limit in the non-archimedean environment.
For the entire collection see [Zbl 1022.00014].
##### MSC:
 14F30 $$p$$-adic cohomology, crystalline cohomology 14J32 Calabi-Yau manifolds (algebro-geometric aspects) 14G20 Local ground fields in algebraic geometry
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