Ordinary Calabi-Yau-3 crystals.

*(English)*Zbl 1063.14023
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, 255-271 (2003).

This is the first of a series of papers which address certain aspects of arithmetic geometry of ordinary Calabi-Yau threefolds and analogies with complex Calabi-Yau threefolds near the large complex structure limit. In particular, this paper discusses ordinary Calabi-Yau-3 crystals, i.e., crystals of the kind that arises as the crystalline cohomology of ordinary Calabi-Yau threefolds in positive characteristics. The main motivation comes from finding a method of proof for the integrality conjectures in mirror symmetry (e.g., Yukawa coupling, prepotential, mirror maps) by reducing them to known results or to easy to check conditions on the crystalline cohomology of families of Calabi-Yau threefolds in positive characteristics.

It is shown that crystals with the properties of crystalline cohomology of ordinary Calabi-Yau threefolds in characteristic \(p>0\) exhibit a remarkable similarity with the well known structure on the cohomology of complex Calabi-Yau threefolds near a boundary point of the moduli space with maximal unipotent local monodromy. In particular, there are canonical coordinates and an analogue of the prepotential of the Yukawa coupling. Moreover new formulas are obtained demonstrating \(p\)-adic analogues of the integrality properties for the canonical coordinates and the prepotential of the Yukawa coupling, which have been observed in the examples of mirror symmetry. For instance, if \(Z\) denotes the prepotential associated to the Yukawa coupling of the quintics in \({\mathbb P}^4\), then for every prime \(p\), the integrality conjecture claims that \[ Z(q)-p^{-3} Z(q^p)\in{\mathbb Z}_p[[q]]\tag{\(*\)} \] where \({\mathbb Z}_p\) is the ring of \(p\)-adic integers. In this paper, it is shown that \(p\)-adic analogues of the integrality conjectures, in particular, a \(p\)-adic analogue of \((*)\), hold true for ordinary Calabi-Yau-3 crystals.

For the entire collection see [Zbl 1022.00014].

It is shown that crystals with the properties of crystalline cohomology of ordinary Calabi-Yau threefolds in characteristic \(p>0\) exhibit a remarkable similarity with the well known structure on the cohomology of complex Calabi-Yau threefolds near a boundary point of the moduli space with maximal unipotent local monodromy. In particular, there are canonical coordinates and an analogue of the prepotential of the Yukawa coupling. Moreover new formulas are obtained demonstrating \(p\)-adic analogues of the integrality properties for the canonical coordinates and the prepotential of the Yukawa coupling, which have been observed in the examples of mirror symmetry. For instance, if \(Z\) denotes the prepotential associated to the Yukawa coupling of the quintics in \({\mathbb P}^4\), then for every prime \(p\), the integrality conjecture claims that \[ Z(q)-p^{-3} Z(q^p)\in{\mathbb Z}_p[[q]]\tag{\(*\)} \] where \({\mathbb Z}_p\) is the ring of \(p\)-adic integers. In this paper, it is shown that \(p\)-adic analogues of the integrality conjectures, in particular, a \(p\)-adic analogue of \((*)\), hold true for ordinary Calabi-Yau-3 crystals.

For the entire collection see [Zbl 1022.00014].

Reviewer: Noriko Yui (Kingston)