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 F_pM 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 F_pM| _X\Z induced by the polarization. We show that the canonical morphism of O_X-modules FpM→j∗(FpM|X\Z)induces an isomorphism between F_pM and the subsheaf of j_∗(F_pM| _X\Z) consisting of sections which are locally L² near Z with respect to h and the standard Lebesgue measure on X. In particular, h extends to a singular hermitian metric on F_pM with minimal extension property.