Disproof of modularity of moduli space of CY 3-folds of double covers of P3 ramified along eight planes in general positions

Gerkmann, Ralf; Mao, Sheng; Zuo, Kang

Disproof of modularity of moduli space of CY 3-folds of double covers of P3 ramified along eight planes in general positions

Authors

Gerkmann, Ralf

Mao, Sheng

Zuo, Kang

Description

We prove that the moduli space of Calabi-Yau 3-folds coming from eight planes of $P^3$ in general positions is not modular. In fact we show the stronger statement that the Zariski closure of the monodromy group is actually the whole $Sp(20,R)$. We construct an interesting submoduli, which we call \emph{hyperelliptic locus}, over which the weight 3 $Q$-Hodge structure is the third wedge product of the weight 1 $Q$-Hodge structure on the corresponding hyperelliptic curve. The non-extendibility of the hyperelliptic locus inside the moduli space of a genuine Shimura subvariety is proved.
27 pages