Topology of algebraic varieties 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.  
 A: 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.
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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:


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*Christian-Oliver Ewald (PhD thesis): "Hochschild Homology and De Rham Cohomology of Stratifolds" http://www.him.uni-bonn.de/fileadmin/user_upload/kreck-phd10.pdf
Or you can use Whitney functions 


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*Bryce Chriestenson and Markus Pflaum: "Whitney functions determine the real homotopy type of a semi-analytic set" http://arxiv.org/pdf/1403.1627v1.pdf
