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.



Leray-Hirsch theorem for Dolbeault cohomology
There's a simples proof of Leray-Hirsch theorem for Dolbeault cohomology in this paper: 
<https://arxiv.org/pdf/1806.11435.pdf>


