The differential-graded algebra of complex-valued smooth differential operators $\mathcal{A}_{X, \mathbb C}^{\bullet}$ on an $n$-dimensional complex manifold $X$ (real dimension $=2n$) is acyclic resolution of the constant sheaf $\mathbb C_{X^{2n}}$ of functions on the underlying real $2n$-dimensional manifold of $X$, i.e. we have a quasi isomorphsim
$\mathbb C_{X^{2n}}\rightarrow\mathcal{A}_{X, \mathbb C}^{\bullet}$. (1)
At the same time, the de Rham complex of holomorphic differential operators $\Omega_X^{\bullet}$ on $X$ defines a resolution for the constant sheaf $\mathbb C_{X^{n}}$ of functions on the $n$-dimensional manifold $X$, i.e. we have a quasi-isomorphism
$\mathbb C_{X^{n}}\rightarrow\Omega_X^{\bullet}$. (2)
Both constant sheaves in $(1)$ and $(2)$ are defined on different spaces so they are not isomorphic. Here is the question I have:
The singular cohomology does not care if $X$ is smooth or holomorphic, it cares only about the topology on $X$. In other words we have $H_{\textrm{sing.}}^{\bullet}(X, \mathbb{C}_{X^{2n}})=H_{\textrm{sing.}}^{\bullet}(X, \mathbb{C}_{X^n})$. Since $X$ is paracompact and locally contractible, we also have $H_{\textrm{sing.}}^{\bullet}(X, \mathbb{C}_{X^{2n}})\cong H^{\bullet}(X, \mathbb C_{X^{2n}})\cong \mathbb{H}^{\bullet}(X, \mathcal{A}_{X, \mathbb C}^{\bullet})$ where the second cohomology is the sheaf cohomology of $\mathbb {C}_{X^{2n}}$ and the last one denotes the hypercohomology of $\mathcal{A}_{X, \mathbb C}^{\bullet}$. Similary, we have $H_{\textrm{sing.}}^{\bullet}(X, \mathbb{C}_{X^{n}})\cong H^{\bullet}(X, \mathbb C_{X^{n}})\cong \mathbb{H}^{\bullet}(X, \Omega_{X}^{\bullet})$. These implies the false conclusion $\mathbb{H}^{\bullet}(X, \mathcal{A}_{X, \mathbb C}^{\bullet})\cong\mathbb{H}^{\bullet}(X, \Omega_X^{\bullet})$. That is imposiible and something must be wrong somewhere but I do not see where. Can someone provide some help here. Thanks in advance.
There is nothing wrong here: the conclusion that $\mathbb{H}^{\bullet}(X, \mathcal{A}_{X, \mathbb C}^{\bullet})\cong\mathbb{H}^{\bullet}(X, \Omega_X^{\bullet})$ is correct. This is in contrast with what you would get if you took cohomology of the chain complex of global sections instead of hypercohomology: in that case using $\mathcal{A}_{X, \mathbb C}^{\bullet}$ would still give you singular cohomology (since the sheaves $\mathcal{A}_{X, \mathbb C}^{\bullet}$ are acyclic) but using $\Omega_X^{\bullet}$ would not in general. In other words, to compute singular cohomology you can take the cohomology of the complex of global smooth complex-valued differential forms (this is essentially just the classical de Rham theorem), whereas if you use holomorphic differential forms you instead need to take hypercohomology of the complex of sheaves instead of just cohomology of global sections (and this is a different and somewhat harder result than the classical de Rham theorem).