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Calabi-Yau operators. (English) Zbl 1405.14027
Ji, Lizhen (ed.) et al., 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 (ISBN 978-1-57146-363-0/pbk). Advanced Lectures in Mathematics (ALM) 42, 401-451 (2018).
Summary: Motivated by mirror symmetry of one-parameter models, an interesting class of Fuchsian differential operators can be singled out, the so-called Calabi-Yau operators, introduced by G. Almkvist and W. Zudilin, AMS/IP Stud. Adv. Math. 38, 481–515 (2006; Zbl 1118.14043)]. They conjecturally determine \(Sp(4)\)-local systems that underly a \(\mathbb{Q}\)-VHS with Hodge numbers \[ h^{30}= h^{21}= h^{12}= h^{03}= 1 \] and in the best cases they make their appearance as Picard-Fuchs operators of families of Calabi-Yau threefolds with \(h^{12}= 1\) and encode the numbers of rational curves on a mirror manifold with \(h^{11}= 1\). We review some of the striking properties of this rich class of operators.
For the entire collection see [Zbl 1398.14003].

14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
14J33 Mirror symmetry (algebro-geometric aspects)
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