Hodge modules and singular hermitian metrics

The purpose of this paper is to study certain notions metric positivity called “minimal extension property” for the lowest nonzero piece in Hodge filtration a module. Let X be complex manifold and let $$\mathcal {M}$$ polarized pure module on with strict support X. $$F_p\mathcal smallest filtration. Assume that smooth outside closed analytic subset Z $$j:X\setminus \hookrightarrow X$$ open embedding. h hermitian {M}|_{X\setminus Z}$$ induced by polarization. We show canonical morphism {O}_X$$ -modules $$\begin{aligned} F_p\mathcal {M}\rightarrow j_{*}(F_p\mathcal Z}) \end{aligned}$$ induces an isomorphism between subsheaf $$j_{*}(F_p\mathcal Z})$$ consisting sections which are locally $$L^2$$ near respect standard Lebesgue measure In particular, extends singular minimal property.