Acta Mathematica

Volume 230 (2023)

Number 2

Pixton’s formula and Abel–Jacobi theory on the Picard stack

Pages: 205 – 319



Younghan Bae (Department of Mathematics, ETH Zürich, Switzerland)

David Holmes (Mathematisch Instituut Leiden, Netherlands)

Rahul Pandharipande (Department of Mathematics, ETH Zürich, Switzerland)

Johannes Schmitt (Institut für Mathematik, University of Zürich, Switzerland)

Rosa Schwarz (Mathematisch Instituut Leiden, Netherlands)


Let $A=(a_1,\ldots,a_n)$ be a vector of integers with $d=\sum_{i=1}^n a_i$. By partial resolution of the classical Abel–Jacobi map, we construct a universal twisted double ramification cycle $\mathsf{DR}^{{\sf op}}_{g,A}$ as an operational Chow class on the Picard stack $\mathfrak{Pic}_{g,n,d}$ of $n$-pointed genus-$g$ curves carrying a degree $d$ line bundle. The method of construction follows the $\log$ (and b-Chow) approach to the standard double ramification cycle with canonical twists on the moduli space of curves [37], [38], [56].

Our main result is a calculation of $\mathsf{DR}^{{\sf op}}_{g,A}$ on the Picard stack $\mathfrak{Pic}_{g,n,d}$ via an appropriate interpretation of Pixton’s formula in the tautological ring. The basic new tool used in the proof is the theory of double ramification cycles for target varieties [42]. The formula on the Picard stack is obtained from [42] for target varieties $\mathbb{CP}^n$ in the limit $n\to \infty$. The result may be viewed as a universal calculation in Abel–Jacobi theory.

As a consequence of the calculation of $\mathsf{DR}^{{\sf op}}_{g,A}$ on the Picard stack $\mathfrak{Pic}_{g,n,d}$, we prove that the fundamental classes of the moduli spaces of twisted meromorphic differentials in $\overline{\mathcal{M}}_{g,n}$ are exactly given by Pixton’s formula (as conjectured in [28, Appendix] and [72]). The comparison result of fundamental classes proven in [40] plays a crucial role in our argument. We also prove the set of relations in the tautological ring of the Picard stack $\mathfrak{Pic}_{g,n,d}$ associated with Pixton’s formula.

Received 5 May 2020

Received revised 11 May 2021

Accepted 16 August 2021

Published 18 July 2023