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.

1more comment