The Hodge–Zucker conjecture for intersection and $L^2$-cohomology

Let $X$ be the locally symmetric variety and let $\mathbb{V}$ be the coefficient variation of Hodge structure considered in Theorem 2.8. The theorem identifies intersection cohomology with $L^2$-cohomology by an isomorphism of $\mathbb{R}$-vector spaces:

$$IH^*(\overline{X},\mathbb{V})\cong H^*_{(2)}(X,\mathbb{V}).$$

Hodge–Zucker conjecture. The isomorphism in Theorem 2.8 is an isomorphism of $\mathbb{R}$-Hodge structures. The question refines Zucker's conjecture by asking whether its cohomological isomorphism is compatible with the natural Hodge structures from mixed Hodge module theory and harmonic analysis. It is known in several cases, including cocompact quotients, certain Hilbert modular varieties and complex balls, and in a range of degrees bounded by the codimension of the singular locus; the general question remains open.