Mixed Hodge structure cohomology of fibration

Question

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.

Answer 1

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 $\mathbb G_m \to \mathbb G_m$).

Answer 2

There has been much research on computing $E$-polynomials of character varieties. You can find a lot of general theory by reading those papers (just do a search for key words).

In particular, the theorem you want is Proposition 2.1 here:

Hodge polynomials of the SL(2,C)-character variety of an elliptic curve with two marked points by Marina Logares, Vicente Muñoz.

I quote:

Suppose that $B$ is connected and $\pi : Z \longrightarrow B$ is an algebraic fibre bundle with fibre $F$ (not necessarily locally trivial in the Zariski topology) and that the action of $\pi_1(B)$ on $H^∗_c(F)$ is trivial. Suppose that $Z, F, B$ are smooth. Then $e(Z) = e(F)e(B).$

The hypotheses hold in particular in the following cases:

• $B$ is irreducible and $\pi$ is locally trivial in the Zariski topology.
• $\pi$ is a principal $G$-bundle with $G$ a connected algebraic group.