I was reading the following [paper](https://eudml.org/doc/286980) which claims to generalize Borel spectral sequence for non-compact Stein fibers. However, I don't understand how the following bundle fits into the picture:


$$
(\mathbb{C}^\times)^2\xrightarrow{\mathbb{C}} \mathbb{T}
$$
$\mathbb{T}$ is a compact torus.
Here $\mathbb{C}$ acts on $(\mathbb{C}^\times)^2$ as a multiplicative subgroup of the form:
$$
\mathbb{C} \times (\mathbb{C}^\times)^2 \ni (z,t\_1,t\_2)\mapsto (e^{iz}\cdot t\_1,e^{z}\cdot t\_2).
$$


If I am not mistaken the fiber is Stein and acyclic. However, Dolbeault cohomology of $\mathbb{T}$ differs drastically from that of $(\mathbb{C}^\times)^2$. Am I missing something?



Borel spectral sequence with non-compact fibers
The second page of the Borel spectral sequence in your example is
$$
E\_2^s=\bigoplus\_{p,q} \mathrm H^{s-q,q}(\mathbf T,\Omega^{p+q-s}(\mathbf C))=\bigoplus\_{p,q} \mathrm H^{s-q,q}(\mathbf T)\otimes\_{\mathbf C}\Omega^{p+q-s}(\mathbf C)
$$
and comprises Dolbeault cohomology of the compact complex torus $\mathbf T$ with values in the infinite-dimensional complex vector spaces $\Omega^{p+q-s}(\mathbf C)$ of all holomorphic $(p+q-s)$-forms on the Stein manifold $\mathbf C$ (infinite-dimensional when $p+q-s=0,1$). That makes it a lot more plausible to converge to the ordinary Dolbeault cohomology of $(\mathbf C^\star)^2$ with values in $\mathbf C$!


