$\quad$Let $\mathcal{Sch}/\mathbb C$ denote the category of schemes over $\mathbb C$. For an arbitrary $X\in\mathcal{Ob}(\mathcal{Sch}/\mathbb C)$, Deligne in his Article defined a polarizable Hodge complex by replacing $X$ by a smooth simplicial scheme $\tilde{X}_\bullet$, and then compactifying by $\tilde{X}_\bullet\rightarrow \bar{\tilde{X}}_\bullet$, such that $\tilde{D}=\bar{\tilde{X}}_\bullet\setminus\tilde{X}_\bullet$ is a simplicial normal crossing divisor. Deligne’s construction then furnished a cosimplicial polarizable Hodge complex, and by taking the corresponding simple complex, one obtains a polarizable Hodge complex $\underline{\mathrm{R}\Gamma}(X,A)\in \mathcal{Ob}(\mathrm{D}_{\mathcal H^p}^b)$. Along this way, one can construct a variant with compact supports $\underline{\mathrm{R}\Gamma}_c(X,A)\in \mathcal{Ob}(\mathrm{D}_{\mathcal H^p}^b)$ for $X\in \mathcal{Ob}(\mathcal{Sch}_\ast/\mathbb C)$ where $\mathcal{Sch}_\ast/\mathbb C$ is category of schemes over $\mathbb C$ and proper morphism. For a closed subscheme $Y\subset X$ there is a distinguished triangle $$…\rightarrow\underline{\mathrm{R}\Gamma}_c(X\setminus Y,A)\rightarrow \underline{\mathrm{R}\Gamma}_c(X,A)\rightarrow \underline{\mathrm{R}\Gamma}_c(Y,A)\rightarrow …$$$\quad$So now we take a further consider, is there a homological counterpart with compact supports and a Borel-Moore version with all standard functoriality properties leads to define the absolute Hodge cohomology of a scheme?
$\quad$For any question about the notations, please check here(1) here(2) here(3)
