$L^2$-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 $IC_X(\mathbb{V})$ via differential forms with locally $L^2$ boundary conditions. When $\mathbb{V}=\mathbb{C}_{X_{\rm reg}}$ 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 $ds^2$ on $X_{\rm reg}$ such that there is a canonical isomorphism $$H^i_{(2)}(X_{\rm reg},ds^2)\simeq IH^i(X),\quad \forall i.$$ Such metric $ds^2$ 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.