Multiplicativity of perverse filtration for Hilbert schemes of fibered surfaces

Let S→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⁎(S[n],Q) for the natural morphism S[n]→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.