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I was reading the following paper 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?

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  • $\begingroup$ Doesn't this action factorize through the diagonal action of $\mathbf C^\star$ on $(\mathbf C^\star)^2$? It seems to me that the quotient is isomorphic to $\mathbf C^\star$ and not to $\mathbf T$. $\endgroup$
    – Johannes Huisman
    Commented Jan 26, 2022 at 16:17
  • $\begingroup$ What is $\Bbb{T}$? $\endgroup$
    – abx
    Commented Jan 26, 2022 at 16:41
  • $\begingroup$ @JohannesHuisman Sorry, yes, I fixed it, I mean there is such action that the quotient is compact. I just wrote down the wrong formula $\endgroup$
    – Grisha Taroyan
    Commented Jan 26, 2022 at 17:05
  • $\begingroup$ @abx compact torus of complex dimension 1 $\endgroup$
    – Grisha Taroyan
    Commented Jan 26, 2022 at 17:09
  • $\begingroup$ Which one? There are many of them (a 1-dimensional family). $\endgroup$
    – abx
    Commented Jan 26, 2022 at 20:25

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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 $\mathbf C$!

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    $\begingroup$ Somehow I thought that the second page is the tensor product of sums of homology groups, not homology and forms. Why is it the same thing? The original result of Borel says something along the lines $E_2^{p,q}=\bigoplus_{p,q} \mathrm H^{s-q,q}(B,\mathbf{H}^{p+q-s}(F)),$ where $\mathbf{H}^{p+q-s}(F)$ is the bundle of Dolbeault cohomology groups of the fibers. The term $E_1,$ however, should look like what you have written down. Maybe the passage to $E_2$ is trickier and we don't get the tensor product like in the case of Kahler fibers $\endgroup$
    – Grisha Taroyan
    Commented Feb 2, 2022 at 8:27
  • $\begingroup$ The complex vector space $\Omega^{p+q-s}(\mathbf C)$ of global holomorphic $(p+q-s)$-forms is the Dolbeault cohomology group $\mathrm H^{p+q-s,0}(\mathbf C)$. The higher $\mathrm H^{p+q-s,i}(\mathbf C)$, $i>0$, vanish since $\mathbf C$ is Stein. Hence, what I wrote down is the $E_2$ page allright. $\endgroup$
    – Johannes Huisman
    Commented Feb 2, 2022 at 11:02

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