New families of Calabi-Yau threefolds without maximal unipotent monodromy

The aim of this paper is to construct families of Calabi-Yau threefolds without boundary points with maximal unipotent monodromy and to describe the variation of their Hodge structures. In particular five families are constructed. In all these cases the variation of the Hodge structures of the Calabi-Yau threefolds is basically the variation of the Hodge structures of a family of curves. This allows us to write explicitly the Picard-Fuchs equation for the one-dimensional families. These Calabi-Yau threefolds are desingularizations of quotients of the product of a (fixed) elliptic curve and a K3 surface admitting an automorphisms of order 4 (with some particular properties). We show that these K3 surfaces admit an isotrivial elliptic fibration.