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In Bott and Tu's Differential forms in algebraic topology there is a proof of Leray-Hirsch for the De Rham cohomology. The theorem is this:

Theorem (Leray-Hirsch): Let $E$ be a fiber bundle over $M$ with fiber $F$. Suppose that $M$ has a finite good cover. If there are global cohomology classes $e_1, > \ldots, e_r$ on $E$ which when restricted to each fiber freely generate the cohomology of the fiber, then $H^*(E)$ is a free module over $H^*(M)$ with basis $\{e_1, \ldots, e_r\}$, i.e. $$ H^*(E) = H^*(M) \otimes \mathbb{R}\{e_1, \ldots, e_r\}.$$

Does the same formula apply for sheaf cohomology in general? Or at least for some "good" sheaves like the sheaf of smooth functions or holomorphic functions? I am asking this because I know that Leray-Hirsch theorem is a very particular case of Leray spectral sequence which is valid for sheaf cohomology in general. But I do not know how to prove that the spectral sequence degenerates at the $E_2$ term.

If the formula applies to the general case, why does the spectral sequence degenerates at the $E_2$ term? References for this results would be greatly appreciated.

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  • $\begingroup$ For proper smooth morphisms of complex varieties there are some technical conditions that guarantee the degeneration. In any case, you will not, in general, get that tensor product unless the $R^q f_{*} \mathscr{F}$ is constant sheaf of $k$-module of finite dimension (I'm assuming that $\mathscr{F}$ is a sheaf of $k$-modules ). The point is that in general the sheaf of differential forms is not acyclic since there's no partition of unity in the algebraic and holomorphic case (that's the whole point of using hypercohomology). $\endgroup$ – user40276 Sep 12 '17 at 15:59
  • $\begingroup$ @user40276 Do you have a reference for this result? I believe that in the case I am working with I have that $R^q f_* \mathcal F$ is constant and finite dimensional. $\endgroup$ – Max Reinhold Jahnke Sep 12 '17 at 17:30
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    $\begingroup$ The degeneration (in some specific cases) which I was talking about is a theorem by Deligne which implies, for instance, the degeneration for proper submersions of Kähler manifolds that restrict on each fiber to a Kähler manifold. The original reference is numdam.org/article/PMIHES_1968__35__107_0.pdf. With a fast search on google, I've found this exposition too math.harvard.edu/~yifei/Deligne_paper.pdf (I don't know if it's good, I haven't read it carefullly). Maybe I can give you a more precise answer if you let me know the case that you're working with. $\endgroup$ – user40276 Sep 14 '17 at 3:34
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There's a simples proof of Leray-Hirsch theorem for Dolbeault cohomology in this paper: https://arxiv.org/pdf/1806.11435.pdf

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