Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Fourier transforms and Abel-Jacobi theory

Younghan Bae, Sam Molcho, Aaron Pixton

TL;DR

The paper develops a canonical Fourier-transformation framework for fine compactified Jacobians over the moduli space of stable curves, resolving the nonextendability of the Poincaré line bundle via log modifications and establishing a derived equivalence (Arinkin kernel). It then connects this Fourier theory to logarithmic Abel–Jacobi theory, showing that the image of resolved Abel–Jacobi sections aligns with the universal double ramification (uniDR) formula and its top-degree part, yielding genus-recursive structure. A central outcome is that pushforwards of divisor monomials on compactified Jacobians can be expressed in terms of twisted DR cycles DR^c_g(b; a), with both the universal and genus-fixed formulations sharing a common recursive pattern. The work also extends the Fourier-transforms to other toroidal abelian fibrations, clarifies the role of residues and Todd classes, and lays groundwork for future analyses of the tautological rings in related moduli problems (e.g., uniDR and higher toroidal fibrations).

Abstract

We relate Fourier transforms between compactified Jacobians over the moduli space of stable curves to logarithmic Abel-Jacobi theory. As an application, we compute the pushforward of divisor monomials on compactified Jacobians in terms of the twisted double ramification cycle formula.

Fourier transforms and Abel-Jacobi theory

TL;DR

The paper develops a canonical Fourier-transformation framework for fine compactified Jacobians over the moduli space of stable curves, resolving the nonextendability of the Poincaré line bundle via log modifications and establishing a derived equivalence (Arinkin kernel). It then connects this Fourier theory to logarithmic Abel–Jacobi theory, showing that the image of resolved Abel–Jacobi sections aligns with the universal double ramification (uniDR) formula and its top-degree part, yielding genus-recursive structure. A central outcome is that pushforwards of divisor monomials on compactified Jacobians can be expressed in terms of twisted DR cycles DR^c_g(b; a), with both the universal and genus-fixed formulations sharing a common recursive pattern. The work also extends the Fourier-transforms to other toroidal abelian fibrations, clarifies the role of residues and Todd classes, and lays groundwork for future analyses of the tautological rings in related moduli problems (e.g., uniDR and higher toroidal fibrations).

Abstract

We relate Fourier transforms between compactified Jacobians over the moduli space of stable curves to logarithmic Abel-Jacobi theory. As an application, we compute the pushforward of divisor monomials on compactified Jacobians in terms of the twisted double ramification cycle formula.

Paper Structure

This paper contains 55 sections, 77 theorems, 361 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Overview
  3. Compactified Jacobians
  4. Twisted double ramification cycle formula
  5. Intersection theory on compactified Jacobians
  6. Logarithmic Poincaré line bundle and Arinkin's kernel
  7. Structure of the universal double ramification cycle formula
  8. Fourier transform and the resolved Abel-Jacobi sections
  9. Other toroidal abelian fibrations
  10. Relation to other work and further discussion
  11. Compactified Jacobians
  12. Compactified Jacobian and logarithmic Picard group
  13. Universal Picard stack and tautological classes
  14. Extending the Poincaré line bundle
  15. Poincaré line bundle
  16. ...and 40 more sections

Key Result

Theorem 1.1

Let $\pi: \overline{J}_{g,n}^\epsilon\to\overline{\mathcal{M}}_{g,n}$ be the compactified Jacobian for a non-degenerate stability condition $\epsilon$. Let $\ell,m,k_2,\ldots, k_n$ be non-negative integers.

Figures (1)

  • Figure 1: The blowup of $\mathbb{P}^1 \times \mathbb{P}^1$ at $(n_1,0),(n_2,\infty)$ and its torus invariant divisors.

Theorems & Definitions (155)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: Theorem \ref{['thm:top']}
  • Theorem 1.4: Theorem \ref{['thm:DR_push']}
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 2.1: MW22
  • Theorem 2.1: MW22
  • Lemma 2.2
  • proof
  • ...and 145 more