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

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.

41 pages. This is a major revision to address errors in lemmas 3.1 and 3.11. The main results are largely unchanged, but the proof of Theorem 1 now works via reduction to a localisation computation, and requires of characteristic zero. We have added invariance properties and a generalisation of a result of Pagani--Ricolfi--van Zelm which were previously in arXiv:2309.00315. Comments welcome!