The Hodge-Double-Ramification conjecture and Mumford's formula on the universal Picard stack

TL;DR

This paper proves the Hodge-DR Conjecture and its logarithmic analogue, providing new formulas for Chern classes on the universal Picard stack, with implications for computing Euler characteristics of differential strata.

Contribution

It establishes the Hodge-DR Conjecture and introduces a Mumford-type formula for the universal jacobian, extending previous work and simplifying proofs.

Findings

Proof of the Hodge-DR Conjecture using localization techniques

Derivation of a Mumford-type formula for the universal jacobian

Shorter proof of a recent result by Holmes et al.

Abstract

The double ramification (DR) cycle associated to a line bundle on a family of curves detects where the line bundle becomes fibrewise-trivial. The Hodge-DR Conjecture proposes a formula for powers of the first Chern class of a natural line bundle on the DR cycle, with a number of applications in the computation of Euler characteristics of strata of differentials. In this paper we prove the conjecture, as well as an analogue for the logarithmic DR cycle. The proof of the former proceeds via reduction to a localisation computation of Fan, Wu and You; the proof of the latter is based on the Thom--Porteous formula, and as a special case gives a shorter proof of a recent result of Holmes, Molcho, Pandharipade, Pixton and Schmitt. Along the way we develop an analogue of Mumford's formula for the Chern character of the universal line bundle on the universal jacobian over the moduli space of twisted curves, generalising work of Mumford, Chiodo, and Pagani--Ricolfi--van Zelm.