Cohomology of fixed point sets of anti-symplectic involutions in the Hilbert scheme of points on a surface (2311.16287v2)
Abstract: Let $S$ be a smooth, quasi-projective complex surface with complex symplectic form $\omega \in H0(S, K_S)$. This determines a symplectic form $\omega_n$ on the Hilbert scheme of points $S{[n]}$ for $n \geq 1$. Let $\tau$ be an anti-symplectic involution of $(S,\omega)$: an order two automorphism of $S$ such that $ \tau*\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 $H*( (S{[n]})\tau; \mathbb{Q})$ in terms of the mixed Hodge structures of $H*( S\tau;\mathbb{Q})$ and of $H*( S / \tau; \mathbb{Q})$. We also classify the connected components of $(S{[n]})\tau$ 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.