$L^2$-representation of Hodge Modules

Abstract

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 $\mathbb{V}={\mathbb{C}}_{X_{\mathrm{reg}}}$ 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 $d{s}^2$ on $X_{\mathrm{reg}}$ such that there is a canonical isomorphism $$H_{(2)}^i(X_{\mathrm{reg}},d{s}^2)\simeq I{H}^i(X),\forall i.$$ Such metric $d{s}^2$ could be Kähler if $X$ is a Kähler space. As an application, we give a differential geometrical proof of the Kähler package of the hypercohomology of pure Hodge modules. We also prove the Kähler version of Kashiwara's conjecture in the absolute case.

58 pages. Comments are welcomed