Mixed Hodge structure on configuration spaces

Let $X$ be a smooth complex projective variety. Let $F(X,n)$ be the configuration space parametrizing $n$ distinct ordered points in $X$. The cohomology groups $H^k(F(X,n),\mathbf Q)$ carry a mixed Hodge structure, by Deligne.

Is there an example where this mixed Hodge structure does not split? In other words, where there is no isomorphism $H^k(F(X,n), \mathbf Q) \cong \bigoplus_{i} \mathrm{Gr}^W_i H^k(F(X,n), \mathbf Q)$? I assume the answer is yes and that an example can be found already when $n=3$ and $X$ is a curve of genus $\geq 2$, but I don't know one.

I would also be interested in the analogous question for $\ell$-adic cohomology and $X$ over (say) a finitely generated field.

Motivation: I was doing some computations about cohomology of configuration spaces which led me to making an optimistic conjecture, which I then realised would imply that all mixed Hodge structures of this form must split. This made me no longer believe the conjecture, but I don't actually have a counterexample.

• see Getzler's paper arxiv.org/abs/alg-geom/9510018 Oct 21, 2014 at 11:28
• I know this paper of Getzler; unfortunately it doesn't answer my question. He only looks at the "Euler characteristic" $\sum_k (-1)^k [H^k_c(F(X,n))]$ in the Grothendieck group of mixed Hodge structures. In particular he never considers questions regarding whether extensions are nontrivial. Oct 21, 2014 at 12:50
• Totaro's paper "Configuration spaces of algebraic varieties" shows that the Leray SS for the inclusion into X^n degenerates after the first non-trivial differential using the mixed Hodge structure. See Theorem 3 math.ucla.edu/~totaro/papers/public_html/config.pdf Oct 21, 2014 at 20:20