Multiplicativity of perverse filtration for Hilbert schemes of fibered surfaces, II

Abstract: Let $S\to C$ be a smooth quasi-projective surface properly fibered onto a smooth curve. We prove that the multiplicativity of the perverse filtration on $H^{*(S{[n]},\mathbb{Q})$} associated with the natural map $S^{[n]}\to C^{(n)}$ implies that $S\to C$ is an elliptic fibration. The converse is also true when $S\to C$ is a Hitchin-type elliptic fibration.