Given a smooth complex projective variety $X$, I know that by the "Hodge Theory" we can compute the dimensions of the cohomology groups of the structural sheaf $\mathcal{O}_X$ by: $$H^p(X,\mathcal{O}_X)\cong H^0(X,\Omega^p) $$ (where $\Omega^p$ is the sheaf of regular $p$-forms) by the more general isomorphism: $$H^p(X,\Omega^q)\cong H^q(X,\Omega^p)$$ I know that all of this comes by harmonic analysis, but has very strong applications to algebraic geometry, as the dimension of the global sections of $\Omega^p$ is far easier to compute.
Now, I wondered if it is was possible to get a hint of that isomorphism just working within the framework of algebraic geometry, computing it directly as cohomology of sheaves. What I was thinking about is this: $$0\to \mathbb{C}\to \Omega^\bullet $$ gives a resolution of the constant sheaf on $X$, and so by tensoring I get a resolution of the structural sheaf: $$ 0\to \mathcal{O}_X\to \mathcal{O}_X\otimes_\mathbb{C}\Omega^\bullet$$ The problem now is that in order to get the cohomology I have to apply the functor of global sections, but the tensor product of sheaves is the sheafification of the presheaf which assigns $U\mapsto \mathcal{O}_X(U)\otimes_\mathbb{C}\Omega^p(U)$ so I really don't know how to work with them (I can just handle this object locally) and this makes difficult to compute kernels and images necessary to get the cohomology groups. Any hints or references to books or notes where I can learn something more are highly accepted!