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

Abstract

Let S → C 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 ] , Q ) H^*(S^{[n]},\mathbb {Q}) associated with the natural map S [ n ] → C ( n ) S^{[n]}\to C^{(n)} implies that S → C S\to C is an elliptic fibration. The converse is also true when S → C S\to C is a Hitchin-type elliptic fibration.