Nuer, Howard J.; Devlin, Patrick A strange family of Calabi-Yau 3-folds. (English) Zbl 1366.14038 Bouchard, Vincent (ed.) et al., String-Math 2014. Proceedings of the conference and satellite workshops, University of Alberta, Edmonton, Alberta, Canada, June 9–13, 2014. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-1992-9/hbk; 978-1-4704-3015-3/ebook). Proceedings of Symposia in Pure Mathematics 93, 245-262 (2016). The paper under review constructs a new family of Calabi–Yau threefolds with Hodge numbers \(h^{11}=31,\,h^{12}=1\) mentioned in [L. A. Borisov and H. J. Nuer, Math. Z. 284, No. 3–4, 853–876 (2016; Zbl 1371.14045)].Consider the four \(4\times 4\) particular matrices with entries in \(\mathbb C\) as sctions \(s_1,\dots, s_4\) of the bundle \(Q\otimes Q\) on \(\mathrm{Gr}(2,4)\) (the Grassmannian of planes in \(\mathbb C^4\)), where \(Q\) is the tautological quotient bundle. Then the family of determinantal varieties in \(\mathrm{Gr}(2,4)\) given by the vanishing of \(\text{det}(s_1,\dots, s_4)\) defines a \(1\)-parameter family of nodal Calabi-Yau threefolds with Hodge numbers \((31,1)\), denoted by \(\tilde{X}\). The Picard-Fuchs differentail equation associated to this family and an MUM (maximally unipotent monodromy) point are determined. These results are used to make predictions about instanton numbers of the conjectured mirror family. The \(A\)-model Yukawa coupling of the mirror is computed and some strange phenomenon that the Yukawa coupling is an even function is observed, as well as a possible explanation of this behavior.Also the monodromy action on \(H^3(\tilde{X},{\mathbb{Q}})\) as well as the conifold period are calculated. These results are used to compute the Euler characteristic of the mirror, which turns out to be \(48\); but this is impossible for a Calabi-Yau threefold with Hodge numbers \((1,31)\). Consequently, no mirror is determined.Furthermore, at a degenerate fiber of the \(1\)-parameter family, a new rigid Calabi-Yau threefold with Hodge numbers \((34,0)\) is constructed.For the entire collection see [Zbl 1343.81006]. Reviewer: Noriko Yui (Kingston) MSC: 14J32 Calabi-Yau manifolds (algebro-geometric aspects) 14J30 \(3\)-folds 14J33 Mirror symmetry (algebro-geometric aspects) 14J28 \(K3\) surfaces and Enriques surfaces Keywords:Picard-Fuchs equation; Yukawa coupling; Calabi-Yau threefold Citations:Zbl 1371.14045 Software:Macaulay2 PDFBibTeX XMLCite \textit{H. J. Nuer} and \textit{P. Devlin}, Proc. Symp. Pure Math. 93, 245--262 (2016; Zbl 1366.14038) Full Text: DOI arXiv