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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.

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  • $\begingroup$ see Getzler's paper arxiv.org/abs/alg-geom/9510018 $\endgroup$
    – guest
    Oct 21, 2014 at 11:28
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    $\begingroup$ 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. $\endgroup$ Oct 21, 2014 at 12:50
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    $\begingroup$ 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 $\endgroup$ Oct 21, 2014 at 20:20

2 Answers 2

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In fact the conclusion in Gorinov's paper seems to be false, see

E. Looijenga, "Torelli group action on the configuration space of a surface", arXiv:2008.10556

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The mixed Hodge structure is in fact always split in this case. This is the main theorem of Alexey Gorinov, "A purity theorem for configuration spaces of smooth compact algebraic varieties", arXiv:1702.08428.

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