Global Kodaira Spencer class and Massey products

We define a new notion of supported global deformation class for a semistable family of complex varieties over a curve $f\text{\hspace{0.17em}⁣}:X\to B$. We use this notion to study when $X$, possibly up to a finite covering, has a generically finite morphism onto a product $B\times Y$ with $Y$ of general type.