$\quad$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](http://publications.ias.edu/sites/default/files/Number19.pdf) 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?




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$\quad$For any question about the notations, please check [here(1)](https://i.stack.imgur.com/S3Y6K.jpg) [here(2)](https://i.stack.imgur.com/GHjDH.jpg) [here(3)](https://i.stack.imgur.com/MKSpa.jpg)



Geometric Interpretation of absolute Hodge cohomology
In fairness, some of the credit for the general construction of absolute cohomology should go to Beilinson (see his *Notes on Absolute cohomology*).
I think you'll find the homology version, that you are asking about, discussed in Janssen's article, *Deligne homology, Hodge D-conjecture, and motives*.


