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Suppose $f:X\to Y$ is a proper smooth morphism of two analytic varieties over $\mathbb C$. Let $\mathbb L$ be a local system on $X$, I want to ask do we have $R^if_*(\mathbb L)\otimes \mathcal O_Y\simeq R^if_*(\mathbb L\otimes \mathcal O_X)$ for any $i\ge 0$? Any reference would be helpful.

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    $\begingroup$ This is already false if $Y$ is a point: cohomology of the structure sheaf is not the $\mathbf{C}$-linearization of singular cohomology. $\endgroup$
    – Satan's Minion
    Commented Nov 13 at 3:47
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    $\begingroup$ You have to define the proper smooth pushforward using a version of the algebraic de Rham complex. This is often presented as part of the theory of Riemann-Hilbert correspondence for D-modules. $\endgroup$
    – Will Sawin
    Commented Nov 13 at 5:25

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