Curvature and relative volume forms

Using metric techniques introduced by Berndtsson, we show a result on the constancy of families dominated by a constant variety and, on the opposite side, results on the strong non-isotriviality of certain families of surfaces with positive index and, in arbitrary dimension, in terms of the complex conjugate of a suitable representative of the Kodaira–Spencer class. We also give a metric interpretation of the liftability of relative volume forms.