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Picard-Fuchs uniformization of modular subvarieties. (English) Zbl 1409.32018
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. Adv. Lect. Math. (ALM) 42, 21-54 (2018).
Summary: In Part I we provide a semi-historical introduction to the Deligne-Mostow multiparameter generalization of hypergeometric functions related to uniformization by complex balls. In Part II we present a wholly new differential-algebraic characterization of Shimura subvarieties in moduli spaces of lattice-polarized K3 surfaces. The two “Parts” are explicitly linked with the example of a family of K3 surfaces whose Picard-Fuchs system is a two-parameter (Appell type) hypergeometric system.
For the entire collection see [Zbl 1398.14003].

##### MSC:
 32Q30 Uniformization of complex manifolds 14J28 $$K3$$ surfaces and Enriques surfaces 14G35 Modular and Shimura varieties 14D20 Algebraic moduli problems, moduli of vector bundles 32G13 Complex-analytic moduli problems 01A55 History of mathematics in the 19th century