Towards refined curve counting on the Enriques surface II: Motivic refinements

TL;DR

This paper investigates motivic invariants of Enriques Calabi-Yau threefolds, proposes a conjecture for Hodge numbers of Jacobian fibrations, and explores their asymptotic behavior and extremal properties.

Contribution

It introduces a conjecture linking perverse Hodge numbers to Betti numbers for Jacobian fibrations on Enriques surfaces, advancing understanding of motivic curve counting.

Findings

Computed motivic Pandharipande-Thomas invariants for Enriques threefolds

Formulated a conjecture relating Hodge and Betti numbers of Jacobian fibrations

Derived asymptotic behavior and posed questions on extremal Hodge numbers

Abstract

We study the motivic Pandharipande-Thomas invariants of the Enriques Calabi-Yau threefolds in fiber curve classes by basic computations and analysis of a wallcrossing formula of Toda. Motivated by our results we conjecture a formula for the perverse Hodge numbers of the compactified Jacobian fibration of linar systems on Enriques surfaces in terms of its Betti numbers. This leads to an asymptotic for said Hodge numbers and raises questions about the behaviour of the extremal Hodge numbers.