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

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.