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].

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].

Reviewer: Noriko Yui (Kingston)

##### MSC:

14F30 | \(p\)-adic cohomology, crystalline cohomology |

14J32 | Calabi-Yau manifolds (algebro-geometric aspects) |

14G20 | Local ground fields in algebraic geometry |