Viehweg’s hyperbolicity conjecture for families with maximal variation

We use the theory of Hodge modules to construct Viehweg–Zuo sheaves on base spaces of families with maximal variation and fibers of general type and, more generally, families whose geometric generic fiber has a good minimal model. Combining this with a result of Campana–Păun, we deduce Viehweg’s hyperbolicity conjecture in this context, namely the fact that the base spaces of such families are of log general type. This is approached as part of a general problem of identifying what spaces can support Hodge theoretic objects with certain positivity properties.