Multiplicativity of Perverse Filtration for Hilbert Schemes of Fibered Surfaces

Let $S\to C$ be a smooth projective surface with numerically trivial canonical bundle fibered onto a curve. We prove the multiplicativity of the perverse filtration with respect to the cup product on $H^{\ast }(S^{[n]},\mathbb{Q})$ for the natural morphism $S^{[n]}\to C^{(n)}$. We also prove the multiplicativity for five families of Hitchin systems obtained in a similar way and compute the perverse numbers of the Hitchin moduli spaces. We show that for small values of $n$ the perverse numbers match the predictions of the numerical version of the de Cataldo-Hausel-Migliorini $P=W$ conjecture and of the conjecture by Hausel, Letellier and Rodriguez-Villegas.