Calabi-Yau manifolds and generic Hodge groups (Q1939227)

Consider a local universal family \(f: \mathcal X \to B \) of Calabi-Yau threefolds with Hodge numbers \(h^{3,0} = h^{2,1} = 1\), so \(B\) is a curve. The corresponding variation of Hodge structures is sometimes called \textit{of mirror quintic type}, e.g. by \textit{M. Green, P. Griffiths} and \textit{M. Kerr} [Contemp. Math. 465, 71--145 (2008; Zbl 1159.14004)]. The aim of the present paper is to determine the generic Hodge group \(\mathrm{Hg}(\mathcal X)\). The main theorem gives a list of three possible groups, two of them are realized concretely by C-Y families, while the remaining case at present is only obtained as the group of a family of structures. The first group is \(\mathrm{Sp}(H^3(X,\mathbb Q),Q)\), the symplectic group associated with the intersection form on \(H^3\); in this case \(\mathrm{Hg}(\mathcal X)\) coincides with the algebraic monodromy group if the monodromy action is infinite. Under the same condition, that the monodromy actions is infinite, then one has the second group iff the monodromy preserves \(F^2\). The third group is closely related to the natural representation of \(\mathrm{SL}_{\mathbb R} (2)\) on \(\mathrm{Sym}^3 ( \mathbb R^2 )\). As the author points out, his techniques depend on deep theories, notably VHS, bounded symmetric domains, Shimura varieties. The results are found by means of intricate computations, some of them quite explicit and instructive.