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

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