Hodge classes associated to 1-parameter families of Calabi-Yau 3-folds

We use $L^2$-Higgs cohomology to determine the Hodge numbers of the parabolic cohomology $H^1(\bar{S},j_{\ast }\text{V})$, where the local system $\text{V}$ arises from the third primitive cohomology of family of Calabi-Yau threefolds over a curve $\bar{S}$. The method gives a way to predict the presence of algebraic 2-cycles in the total space of the family and is applied to some examples.