Topological information via cohomology of sheaves

Question

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?

Answer 1

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 $\pi_1(X)$).

Answer 2

Suppose $X$ is compact. For simplicity, first assume that $\mathcal F$ is a holomorphic verctor bundle. Then we can organize the numbers $h^q=dimH^q(X, \mathcal O(\mathcal F))$ as follows: $$\chi(X,\mathcal F):=\sum_{q=0}^{dim X}(-1)^qh^q.$$ Then by Hirzebruch-Riemann-Roch theorem: $$\chi(X, \mathcal F)=\int_X ch(\mathcal F)td(X),$$ where $ch(\mathcal F)$ is the chern character of $\mathcal F$ and $td(X)$ is the Todd class of the tangent bundle of $X$, they are topological invariants of $\mathcal F$ and $TX$ respectively. Now if you want to consider $H^q(X, \Omega^p\otimes \mathcal F)$, you just fix $p$ and take sum of dimesions of the cohomology over $q$, then you get the above formula again, with $\mathcal F$ replaced by $\Omega^p\otimes \mathcal F$. For general cohorent analytic sheaf $\mathcal F$, it is well known that $\mathcal F$ can be resolved by a exact sequence of holomorphic vector bundles, then by using formal minus operation on vector bundles and splitting property of chern characters, you can do the same thing as above.