Title: Hodge sheaves for singular families

Abstract: This is a report on joint work with Behrouz Taji. Given a flat projective morphism f: X→B of complex varieties, assuming that B is smooth, we construct a functorial system of reflexive Hodge sheaves on B. If in addition, X is also smooth then this system gives an extension of the Hodge bundle underlying the VHS of the smooth locus of f. This in turn provides a criterion that all VHSs of geometric origin must satisfy. As an independent application we prove a singular version of Viehweg's conjecture about base spaces of families of maximal variation.