If $X$ is a compact Kahler manifold, then Hodge theory says that its cohomology decomposes as a direct sum

$$ H^{p+q}(X,\mathbb C) = \bigoplus\_{p,q} H^{p,q}(X,\mathbb C) $$

where $H^{p,q}(X,\mathbb C) = H^q(X,\Omega\_X^p)$ are the Dolbeault cohomology groups and $H^{p,q}$ and $H^{q,p}$ are conjugate isomorphic. One can prove that the same holds for a Fujiki manifold, i.e. a complex manifold bimeromorphic to a Kahler manifold.

**Q:** What happens for compact complex spaces?

Here "complex space" should mean singular and non-reduced. Is there a notion of a "Kahler" complex space, on which some predictions of Hodge theory hold? The question should make sense, as we can define $H^{p+q}(X,\mathbb C)$ topologically and the $H^q(X,\Omega\_X^p)$ exist for a complex space $X$, but do we even know that the latter are subgroups of the former for some special spaces $X$?

One can imagine naively defining such spaces as those which admit a Fujiki desingularisation (Kahler to begin with, but desingularisations are not unique so we need to compare them somehow), and then hoping that the "upstairs" decomposition induces one "downstairs". Of course, if it were that simple someone would have done it ages ago.