Calabi-Yau manifolds realizing symplectically rigid monodromy tuples
This work is motivated by the study of mirror symmetry and Calabi-Yau manifolds, i.e., compact Kaehler manifolds with trivial canonical bundle. The \(K3\) surfaces are the only simply connected Calabi-Yau surfaces manifolds. The classification of Calabi-Yau threefolds remains an open problem. Systematic computer searches have given us a better idea of the landscape of Calabi-Yau threefolds (see [\textit{S. B. Johnson} and \textit{W. Taylor}, J. High Energy Phys. 2014, No. 10, Paper No. 023, 42 p. (2014; Zbl 1333.81384); \textit{W. Taylor}, J. High Energy Phys. 2012, No. 8, Paper No. 32, 18 p. (2012; Zbl 1397.14048)]). In particular they have showed the importance of elliptic fibrations on Calabi-Yau threefolds. Unfortunately, it has also been revealed that it is generally quite difficult to construct examples of families of Calabi-Yau threefolds with small Hodge number \(h^{2,1}\) by specialization of multi-parameter families. The mathematical study of mirror symmetry essentially began with the example of the quintic and mirror quintic family of \textit{Ph. Candelas} et al. [``An exactly soluble superconformal theory from a mirror pair of Calabi-Yau manifolds'', Phys. Lett. B 258, No. 1--2, 118--126 (1991; \url{doi:10.1016/0370-2693(91)91218-K})]. The quintic family is a generic quintic hypersurface in \(\mathbb P^5\), e.g., the Fermat quantic \(X^5_0+\cdot +X^5_4 = 0\) ; which is a Calabi-Yau threefold with Hodge numbers \(h^{1,1} = 1\) and \(h^{2,1} = 101\). This family is a one-dimensional family of Calabi-Yau threefolds defined over the base \(\mathbb P^1{0,1,\infty}\) and has exactly three singular fibers: the large complex structure limit, the Gepner point, and the stacky point. Moreover this family gives rise to a variation of Hodge structure and an associated Picard-Fuchs differential equation. In turn, the solutions to the differential equation determine this variation of Hodge structure. The Picard-Fuchs equations for other families of Calabi-Yau threefolds were constructed in [\textit{V. V. Batyrev} et al., Nucl. Phys., B 514, No. 3, 640--666 (1998; Zbl 0896.14025)] and [\textit{V. V. Batyrev} and \textit{D. van Straten}, Commun. Math. Phys. 168, No. 3, 493--533 (1995; Zbl 0843.14016)]. It lead to a general denition of a Calabi-Yau differential operator by Almkvist, van Enckevort, van Straten and Zudilin, rooted purely in the theory of differential operators; see [\textit{G. Almkvist} et al., ``Tables of Calabi-Yau equations'', \url{arxiv:math/0507430}] and [\textit{D. van Straten}, in: Uniformization, Riemann-Hilbert correspondence, Calabi-Yau manifolds and Picard-Fuchs equations. Based on the conference, Institute Mittag-Leffler, Stockholm, Sweden, July 13--18, 2015. Somerville, MA: International Press; Beijing: Higher Education Press. 401--451 (2018; Zbl 1405.14027)]. A large number of these are known today but they are mostly found via computer searches. The authors devise an iterative geometric construction that produces explicit families of elliptically fibered Calabi-Yau manifolds whose Picard-Fuchs operators realize a big class of Calabi-Yau differential operators, including the so-called symplectically rigid operators, but also non-rigid operators. They considers elliptic fibrations with sections, so-called Jacobian elliptic fibrations. These families are then presented as Weierstrass models.
scientific article