Hodge modules and singular hermitian metrics

The purpose of this paper is to study certain notions of metric positivity called "minimal extension property" for the lowest nonzero piece in the Hodge filtration of a Hodge module. Let X be a complex manifold and let M be a polarized pure Hodge module on X with strict support X. Let FpM be the smallest nonzero piece in the Hodge filtration. Assume that M is smooth outside a closed analytic subset Z and let j : X \ Z -> X be the open embedding. Let h be the smooth hermitian metric on FpM|X\Z induced by the polarization. We show that the canonical morphism of OX-modules FpM -> j*(FpM vertical bar X\Z) induces an isomorphism between FpM and the subsheaf of j*(FpM vertical bar X\Z) consisting of sections which are locally L-2 near Z with respect to h and the standard Lebesgue measure on X. In particular, h extends to a singular hermitian metric on FpM with minimal extension property.