Pixton's formula and Abel-Jacobi theory on the Picard stack

Abstract

Let $A=(a_1,\dots ,a_n)$ 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 ${\mathsf{DR}}_{g,A}^{\mathsf{op}}$ 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 [arXiv:1707.02261, arXiv:1711.10341, arXiv:1708.04471]. Our main result is a calculation of ${\mathsf{DR}}_{g,A}^{\mathsf{op}}$ 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 [arXiv:1812.10136]. The formula on the Picard stack is obtained from [arXiv:1812.10136] 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 ${\mathsf{DR}}_{g,A}^{\mathsf{op}}$ 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 the appendix to [arXiv:1508.07940] and in [arXiv:1607.08429]). The comparison result of fundamental classes proven in [arXiv:1909.11981] 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 to Pixton's formula.

arXiv comment122 pages. v2: This is the Author Accepted Manuscript, to appear in Acta Mathematica (open access). Main change since v1 is addition of Section 6.2 discussing a refinement of the definition (due to Marcus and Wise) of logarithmic rubber maps. Comments still very welcome

References in corpus (20)

• Double ramification cycles on the moduli spaces of curves
• Compactification of strata of abelian differentials
• Gromov-Witten theory and Noether-Lefschetz theory
• Normal Functions and the Geometry of Moduli Spaces of Curves
• Parabolic sheaves on logarithmic schemes
• Characteristic classes in the Chow ring
• Integrals of psi-classes over double ramification cycles
• Holomorphic anomaly equations and the Igusa cusp form conjecture
• admcycles -- a Sage package for calculations in the tautological ring of the moduli space of stable curves
• Constructions of nontautological classes on moduli spaces of curves
• The moduli space of multi-scale differentials
• Pixton's double ramification cycle relations
• Stable maps to rational curves and the relative Jacobian
• Relations among tautological classes revisited
• Extending the Double Ramification Cycle using Jacobians
• Infinitesimal structure of the pluricanonical double ramification locus
• A generalization of the double ramification cycle via log-geometry
• Powers of the theta divisor and relations in the tautological ring
• Higher genus relative and orbifold Gromov-Witten invariants of curves
• Volumes of moduli spaces of flat surfaces

Cited by in corpus (1)

• A case study of intersections on blowups of the moduli of curves