  We study period maps for families of $K3$ surfaces those are given by anti
canonical divisors of toric varieties coming from reflexive polytopes $P_2,
P_4, P_5$ and $P_r$. We obtain systems of period differential equations for
these families. Moreover, in the case $P_4$, we determine the projective
monodromy group of the period map. This group is explicitly related with the
Hilbert modular group for $\mathbb{Q}(\sqrt{5})$.
