Cohomology of fixed point sets of anti-symplectic involutions in the Hilbert scheme of points on a surface

Let $S$ be a smooth, quasi-projective complex surface with complex symplectic form $\omega \in H^0(S,K_S)$. This determines a symplectic form ${\omega }_n$ on the Hilbert scheme of points $S^{[n]}$ for $n\ge 1$. Let $\tau$ be an anti-symplectic involution of $(S,\omega )$: an order two automorphism of $S$ such that ${\tau }^{\ast }\omega =-\omega$. Then $\tau$ induces an anti-symplectic involution on $(S^{[n]},{\omega }_n)$ and the fixed point set $(S^{[n]}{)}^{\tau }$ is a smooth Lagrangian subvariety of $S^{[n]}$. In this paper, we calculate the mixed Hodge structure of $H^{\ast }((S^{[n]}{)}^{\tau };\mathbb{Q})$ in terms of the mixed Hodge structures of $H^{\ast }(S^{\tau };\mathbb{Q})$ and of $H^{\ast }(S\mathrm{/}\tau ;\mathbb{Q})$. We also classify the connected components of $(S^{[n]}{)}^{\tau }$ and determine their mixed Hodge structures. Our results apply more generally whenever $S$ is a smooth quasi-projective surface, and $\tau$ is an involution of $S$ for which $S^{\tau }$ is a curve.