  We give a completion of the period map associated to a variation of polarized
Hodge structure arising from a 2-dimensional geometric family that has Hodge
type (1,2,2,1). This is the second known example of a completion of a period
map that is higher than one dimensional with non-hermitian symmetric
Mumford-Tate domain. The completion technique we use is the theory of Kato-Usui
spaces, while the calculation of monodromy matrices is done with the help of
mirror symmetry.
