On extensions of PVMHS and mixed hodge modules

We employ the techniques of mixed Hodge modules in order to answer some questions on extension of mixed Hodge structures. Specifically a theorem of M. Saito tells that, the mixed Hodge modules on a complex algebraic manifold X, correspond to polarized variation of mixed Hodge structures on Zariski open dense subsets of X. In this article we concern with the minimal extension of MHM or PVMHS related to this criteria. In [26] we studied the extension of VMHS associated to isolated hypersurface singularities. This article generalizes some of the results there to the admissible VMHS on open dense submanifolds. Some applications to the Neron models of Hodge structures are also given. A short discussion on abelian positivity in the positive characteristic and of height pairing on arithmetic varieties have been included.