I have a question about the Morgan's paper ``The algebraic topology of smooth complex varieties''. Let $\Bbbk=\mathbb{C}$. Given a smooth non singular variety $X$ with normal crossing divisor $D$, the sheaf logarithmic forms $\mathcal{A}_{DR}^{\bullet}(\log(D))$ is defined as follows: for an open set $U$, a form in $\mathcal{A}_{DR}^{\bullet}(\operatorname{Log}(D))|_{U}$ can be written as

$w=\sum w^{J}\frac{dz_{1}}{z_{1}}\cdots \frac{dz_{j}}{z_{j}}$

where $w^{J}$ is a smooth form on $U$. Let ${A}_{DR}^{\bullet}(\operatorname{Log}(D))$ be the differential graded algebra denoting the global sections of $\mathcal{A}^{\bullet}_{DR}(\operatorname{Log}(D))$. Let $A_{DR}^{\bullet}(X-D)$ be differential graded algebra of the ordinary complex smooth differential forms on $X-D$. Here my question: at page 154 theorem 3.3 it is written that

$(*)$"the inclusion ${A}_{DR}^{\bullet}(log(D))\hookrightarrow A_{DR}^{\bullet}(X-D) $ induces an isomorphism in cohomology"

I know that this statement is true in sheaf terms, i.e. the inclusion induces an isomorphism at the level of the Hypercohomology of the two sheaves, but in seems to me that the statements here are **really** about differential graded algebras.

My problem is the following: Consider a torus $T=\mathbb{C}/\mathbb{Z}^{2}$ with coordinate $z=r+is$ and the obvious group action given by translations. Consider ${A}_{DR}^{\bullet}(T)$. Then the first cohomology group is generated by $dr$ and $ds$ and the second cohomology group is generated by $dr \, ds$. Let $D=0\subset T$ considered as a normal crossing divisor. The cohomology of ${A}_{DR}^{\bullet}(\log(D))$ is generated in degree $1$ by $dr$ and $ds$ and in degree $2$ we should have that $dr \, ds$ is exact. But $A_{DR}(\log(D))^{0}=A^{0}_{DR}(T)$ by definition and hence $drds$ is not exact. Where is my mistake?

Here is the link for the paper: http://www.numdam.org/item/PMIHES_1978__48__137_0