On the global moduli of Calabi-Yau threefolds

In this note we initiate a program to obtain global descriptions of Calabi-Yau moduli spaces, to calculate their Picard group, and to identify within that group the Hodge line bundle, and the closely-related Bagger-Witten line bundle. We do this here for several Calabi-Yaus obtained in [DW09] as crepant resolutions of the orbifold quotient of the product of three elliptic curves. In particular we verify in these cases a recent claim of [GHKSST16] by noting that a power of the Hodge line bundle is trivial -- even though in most of these cases the Picard group is infinite.