Let $X$ be a smooth complex algebraic variety. From Deligne's work, we know that the have a Mixed Hodge structure over its (rational) compactly supported cohomology $H^{\*}\_c(X,\mathbb{Q})$. With this, one can define the Euler-Hodge polynomial $$E(X,x,y)=\sum\_{p,q}e\_{p,q}x^py^q$$ where $$e\_{p,q}=\sum\_i (-1)^{i} dim Gr^W\_{p+q} Gr\_p^F H^i\_c(X,\mathbb{Q}) $$.


Let us suppose we have another complex smooth algebraic variety $P$ with an algebraic map $f:P \to X$ which is a fiber bundle with fiber $F$. I would like to know if there is any result that ensures us that $$E(X,x,y)E(F,x,y)=E(P,x,y).$$


This should somehow be thought of as a generalization of the similar equation which is true for Euler characteristic.



Mixed Hodge structure cohomology of fibration
If $\pi\_1(X)=0$, or more generally if the fundamental group of $X$ acts trivially on the compact support cohomology of the fiber, this is true because of the Leray spectral sequence. Otherwise it is almost never true (a simple counterexample was given by EBz in the comments: the squaring map $\mathbb G\_m \to \mathbb G\_m$).


