Complex multiplication, Griffiths-Yukawa couplings, and rigidity for families of hypersurfaces

Let $\mathcal {M}_{d,n}$ be the moduli stack of hypersurfaces $X \subset \mathbb {P}^n$ of degree $d\geq n+1$, and let $\mathcal M_{d,n}^{(1)}$ be the sub-stack, parameterizing hypersurfaces obtained as a $d$-fold cyclic covering of $\mathbb {P}^{n-1}$ ramified over a hypersurface of degree $d$. Iterating this construction, one obtains $\mathcal {M}_{d,n}^{(\nu )}$. We show that $\mathcal {M}_{d,n}^{(1)}$ is rigid in $\mathcal {M}_{d,n}$, although for $d<2n$ the Griffiths-Yukawa coupling degenerates. However, for all $d\geq n+1$ the sub-stack $\mathcal {M}^{(2)}_{d,n}$ deforms. We calculate the exact length of the Griffiths-Yukawa coupling over $\mathcal {M}_{d,n}^{(\nu )}$, and we construct a $4$-dimensional family of quintic hypersurfaces $g:\mathcal {Z}\to T$ in $\mathbb {P}^4$, and a dense set of points $t$ in $T$, such that $g^{-1}(t)$ has complex multiplication.