Hodge modules and singular hermitian metrics

Abstract

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 $\mathcal {M}$ be a polarized pure Hodge module on X with strict support X. Let $F_p\mathcal {M}$ be the smallest nonzero piece in the Hodge filtration. Assume that $\mathcal {M}$ is smooth outside a closed analytic subset Z and let $j:X\setminus Z \hookrightarrow X$ be the open embedding. Let h be the smooth hermitian metric on $F_p\mathcal {M}|_{X\setminus Z}$ induced by the polarization. We show that the canonical morphism of $\mathcal {O}_X$-modules

$$\begin{aligned} F_p\mathcal {M}\rightarrow j_{*}(F_p\mathcal {M}|_{X\setminus Z}) \end{aligned}$$

induces an isomorphism between $F_p\mathcal {M}$ and the subsheaf of $j_{*}(F_p\mathcal {M}|_{X\setminus Z})$ 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 $F_p\mathcal {M}$ with minimal extension property.