  This proceedings note introduces aspects of the authors' work relating mirror
symmetry and integral variations of Hodge structure. The emphasis is on their
classification of the integral variations of Hodge structure which can underly
families of Calabi-Yau threefolds over the thrice-punctured sphere with b^3 =
4, or equivalently h^{2,1} = 1, and the related issues of geometric realization
of these variations. The presentation parallels that of the first author's talk
at the BIRS workshop.
