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

TL;DR

This paper establishes a link between the multiplicativity of the perverse filtration on Hilbert schemes of fibered surfaces and the elliptic nature of the fibration, providing a characterization of elliptic fibrations via perverse filtrations.

Contribution

It proves that multiplicativity of the perverse filtration characterizes elliptic fibrations for fibered surfaces, and conversely, that certain elliptic fibrations exhibit this multiplicativity property.

Findings

Multiplicativity of the perverse filtration implies the surface is elliptic.

The converse holds for Hitchin-type elliptic fibrations.

Provides a new criterion for identifying elliptic fibrations.

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.