On a projective smooth variety $X$ over complex numbers (or rather compact Kahler) we have a specific set of sheaves, namely sheaves of holomorphic forms ${\mathcal \Omega}^p$ of various degrees. The cohomology $H^q(X, \Omega^p)$ of these sheaves "fit" together via Hodge decomposition into cohomology groups of our variety (manifold), which are of purely topological origin. 


Assume we have a sheaf $\mathcal F$ on $X$ (say, coherent, although I don't really know how relevant this is). 


Can we "fit" together groups $H^q(\Omega^p\otimes \mathcal F)$ "analogously" to Hodge decomposition (where we set $\mathcal F=O\_X$) to get a purely topologically defined object
(maybe originating now not from $X$ but from another variety)?


If sheaves $\Omega^p$ do not work for arbitrary such $\mathcal F$, can we find their respective analogs $\mathcal \Omega^p\_{F}$ to fit the corresponding groups together as it is suggested above?



Topological information via cohomology of sheaves
I think the situation you are looking for is when $\mathcal F$ is a local system. If $F$ is a locally constant sheaf on $X$, say of $\mathbf C$-vector spaces, then you can put $\mathcal F = F \otimes\_\mathbf{C} \mathcal O$ to get a holomorphic vector bundle, which is canonically equipped with a flat connection 
$$ \nabla : \mathcal F \to \Omega^1 \otimes \mathcal F$$
such that $F = \mathrm{ker}(\nabla)$.
Flatness means that when you extend $\nabla$ to a map $\Omega^d \otimes \mathcal F \to \Omega^{d+1}\otimes \mathcal F$ then $\nabla \circ \nabla = 0$, so you get a complex of sheaves $\Omega^\bullet \otimes \mathcal F$. The hypercohomology $\mathbb H^i(X,\Omega^\bullet \otimes \mathcal F)$ is isomorphic to the cohomology $H^i(X,F)$, which is a purely topological object ($F$ is just a complex representation of $\pi\_1(X)$). 


