cohomology of configuration space of punctured variety

Question

Given a smooth projective variety $X$ of dimension $l$, we denote with $F(X,n)$ the configuration space of points $$ F(X,n):=\{(x_{1}, \dots, x_{n})\in X^{n}\: : \: x_{i}\neq x_{j}\text{ for each }i,j \} $$ In https://www.jstor.org/stable/2946581?seq=1#page_scan_tab_contents there is an explicit rational dg algebra $E(n)$ such that $H^{\bullet}(E(n))\cong H^{\bullet}(X, \mathbb{Q})$ (a model). In particular , as an algebra, $E(n)$ is isomorphic to a free $H^{\bullet}(X^{n})$ algebra with generators $G_{ab}$ for $1\leq a,b \leq l$, modulo some easy relations. The same is true for the differential. Now, choose $k$ distinct points $y_{1}, \dots y_{k}$ on $X$. We consider the configuration space of $X-\{y_{1}, \dots , y_{k}\}$

$$F(X, n; y_{1}, \dots , y_{k}):=F(X-\{y_{1}, \dots , y_{k}\},n)$$

Here my question: do you know a model $E(n, k )$ that compute the rational cohomology of $F(X, n; y_{1}, \dots , y_{k})$? What are generators, relations and the differential?

I think that such a model can be obtained from $E(n)$ by the fact that $F(X, n; y_{1}, \dots , y_{k})$ is isomoprhic to the fiber at the point $(y_{1}, \dots, y_{k})$ of the projection $$ F(n+k,X)\rightarrow F(k, X) ,$$ but I wonder if there is something in the literature. I am interested to the case $dimX=1$.

Edit: I don't know if this is an open problem. I am interested to find a 1-minimal model for the case $dimX=1$, $k=1$. Is there a way procedure to compute $H^{1}( F(n, X;y) $ and the subspace $V\subset H^{2}( F(n, X;y)$ generated by $H^{1}( F(n, X;y) $ via the wedge product?

Answer 1

If you read Totaro's paper "Configuration spaces of algebraic varieties" he derives the same cdga calculating the cohomology of $F(X,n)$ as Kriz, but in a different way, via the Leray spectral sequence for $F(X,n) \to X^n$. First he calculates what the first nontrivial page of the spectral sequence looks like for an arbitrary oriented manifold, and he finds that it's given precisely by this expression $$H^\ast(X^n,\mathbf Q)[G_{ab}]/\text{relations.}$$ Then he shows that if $X$ is smooth projective algebraic, the spectral sequence degenerates after the first differential using a weight argument: no differential after the first one can be compatible with the weights in the natural mixed Hodge structure on the spectral sequence. The argument works more generally if $H^i(F(X,n),\mathbf Q)$ is pure of weight $i$ for all $i$ --- in particular, it works for a once-punctured smooth projective variety. So for $k=1$, the exact same cdga works to compute the cohomology of $F(X,n)$. No assumption on the 1-connectedness of $X$ is necessary.

Also. You may find the paper "Koszul dg-algebras arising from configuration spaces" by Bezrukavnikov useful.

Re your last question, let $X$ be a once punctured Riemann surface with first cohomology group $V$. Then $H^1(F(X,n),\mathbf Q) \cong V^{\oplus n}$, i.e. the map $F(X,n) \to X^n$ induces an isomorphism on $H^1$. The image of $H^1(F(X,n)) \otimes H^1(F(X,n)) \to H^2(F(X,n))$ is the direct sum of $\binom n 2$ copies of $V_{2,0} \oplus V_{1,1}$, where these denote representations of $\mathrm{Sp}(V)$. Specifically, $V_{2,0}$ is the symmetric square, and $V_{1,1}$ is the "primitive part" of the exterior square (the orthogonal complement of the class of the symplectic form).

Answer 2

The question is addressed by Berceanu, Markl, and Papadima in Multiplicative models for configuration spaces of algebraic varieties, Topology 44 (2005), no. 2, 415–440.