$L^2$-representation of Hodge Modules

Over an arbitrary compact complex space or an arbitrary germ of complex space $X$, we provide fine resolutions of pure Hodge modules with strict supports $I{C}_X(\mathbb{V})$ via differential forms with locally $L^2$ boundary conditions. When $\mathbb{V}={\mathbb{C}}_{X_{\mathrm{r}\mathrm{e}\mathrm{g}}}$ is the trivial variation of Hodge structure, we give a solution to a Cheeger-Goresky-MacPherson type conjecture: For any compact complex space $X$, there is a complete hermitian metric $d{s}^2$ on $X_{\mathrm{r}\mathrm{e}\mathrm{g}}$ such that there is a canonical isomorphism $$H_{(2)}^i(X_{\mathrm{r}\mathrm{e}\mathrm{g}},d{s}^2)\simeq I{H}^i(X),\mathrm{\forall }i\mathrm{.}$$ Such metric $d{s}^2$ could be K"ahler if $X$ is a K"ahler space. As an application, we give a differential geometrical proof of the K"ahler package of the hypercohomology of pure Hodge modules. We also prove the K"ahler version of Kashiwara’s conjecture in the absolute case.