  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.
