A question about the logarithmic complex and Morgan's paper

Question

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

Answer 1

For the first part of the question: what is well-known is that the cohomology of the open variety $X\setminus D$ can be computed as the hypercohomology of the holomorphic logarithmic complex $\Omega_X^\bullet(\log D)$. This is well explained in Voisin's book Hodge Theory and Complex Algebraic Geometry, corollary 8.19. But the above sheaf $\mathcal{A}^\bullet(\log D)$ is an acyclic resolution of $\Omega_X^\bullet(\log D)$ because it is a sheaf of modules overs $\mathcal{C}^\infty$ functions on $X$. This is stated in Deligne Hodge II, 3.2.3 b. Hence the hypercohomology of $\Omega_X^\bullet(\log D)$ can be computed as the cohomology of the global sections of $\mathcal{A}^\bullet(\log D)$. Again in your assertion $(*)$ that's because your two sheaves are acyclic and compute both the cohomology of $X\setminus D$ that the map is an isomorphism at the level of cohomology of the global sections. This is a general fact about holomogical algebra, see also Voisin's book corollary 8.9 and proposition 8.12.

For your second part. Your computation of $A^\bullet(\log D)$ is correct and agrees with the cohomology of the open manifold $T\setminus 0$ (which you seem to confuse with the cohomology of $T$), in particular $H^2(T \setminus 0)=0$. One way to see it is that $T\setminus 0$ deformation retracts onto a wedge sum of two circles; or invoke Poincaré duality for oriented non-compact manifolds, which implies that the top cohomology group is zero.

Answer 2

EDIT: So my attempt at manipulating forms was a failure, so let's just go by general non-sense.

These are CDGA's in the $j^{-1}\mathcal{O}_{X}$-module category (after pulling back $A^\bullet_X(\log D))$ ), so we really just need to check that the quasi-isomorphism preserves the multiplication structure. This can be checked locally, but in local coordinates it's more-or-less obvious that the wedge of logarithmic forms maps to the wedge of the forms restricted to $X \setminus D$.