Let $X$ be a projective variety (lets say normal and irreducible) with the topology coming from being a subspace of $\mathbb{P}^N$ (and not the Zariski topology). Surely one can then define the singular cohomology groups. My question is whether one can also make sense of the Cech cohomology groups $H^*(X,\mathbb{C})$ for the sheaf of locally constant $\mathbb{C}$-valued functions, and if the two cohomology groups agree, like would be the case if $X$ were a complex manifold. Is there also a notion of De Rahm cohomology where we look at forms in some suitable Sobolev space?
I apologize for the rather vague/soft question, but I was not able to find any references online. So it would also be great if someone could point out references for this sort of thing.
Best Answer
Regarding your question on De Rham cohomology there are several approches to realize a De Rham complex that computes singular cohomology.
A. In Algebraic geometry.
You should look at R. Hartshorne "Algebraic De Rham cohomology" manuscripta math. 7, 125-140 (1972). It is a research announcement and survey on the cohomology of algebraic De Rham forms on algebraic varieties, details are published in the Publications of IHES (1975).
In particular for a scheme $Y$ of finite type over a characteristic zero field $k$, he defines its algebraic De Rham cohomology.
1) You embed $Y$ as a closed subscheme of a smooth scheme $X$.
2) You consider $\Omega^*$ the complex of sheaves of regular differential forms on $X$ over $k$.
3) You take $\hat{X}$ the formal completion of $X$ along $Y$: $$\hat{Y}=\bigcup_n Y(n)$$ where $Y(n)$ is the infinitesimal neighbourhood of order $n$ of $Y$ in $X$ and consider $\hat{\Omega}^*$ the completion of $\Omega^*$.
4) You define $H^*_{DR}(Y)$ as the hypercohomology of the complex $\hat{\Omega}^*$ on the formal scheme $\hat{X}$.
Then (theorem 1.6 of this paper) when $Y$ is a scheme of finite type over $k=\mathbb{C}$ we have a natural isomorphism $$H^i_{DR}(Y)\cong H^i(Y^{an},\mathbb{C})$$ where $Y^{an}$ is the corresponding complex analytic space and $H^i(-,\mathbb{C})$ is the singular cohomology.
...............................
B. As stratified spaces
You can use the fact that a complex algebraic variety is stratified, for example it is a stratifold in the sense of M. Kreck, then you have a notion of De Rham complex that computes singular cohomology with real coefficients:
Or you can use Whitney functions