Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An Abel-Jacobi theorem for metrized complexes of Riemann surfaces

Maximilian C. E. Hofmann, Martin Ulirsch

TL;DR

The paper proves an Abel-Jacobi theorem for metrized complexes of Riemann surfaces (mCRS), unifying the classical Abel-Jacobi theorem with its tropical analogue in hybrid geometric settings. By introducing the Jacobian $\mathrm{Jac}(\mathfrak{X})=\Omega^*(\mathfrak{X})/H_1(\mathfrak{X})$ and an Abel-Jacobi map $A_{\mathfrak{X},0}$, it provides a canonical isomorphism $\mathrm{Jac}(\mathfrak{X}) \cong \mathrm{Div}_0(\mathfrak{X})/\mathrm{PDiv}(\mathfrak{X})$ and shows that degree-zero divisors are principal precisely when they map to zero under $A_{\mathfrak{X},0}$. The result interpolates between $\mathrm{Jac}(X_v)$ and $\mathrm{Jac}(\Gamma)$, recovers the classical Abel-Jacobi theorem for Riemann surfaces and the tropical theorem for metric graphs, and provides a robust framework for limit linear series in hybrid spaces. The methods combine Abel-Jacobi theory for components and graphs with a snake-lemma argument to derive the global statement on mCRS.

Abstract

Motivated by the recent surge of interest in the geometry of hybrid spaces, we prove an Abel-Jacobi theorem for a metrized complex of Riemann surfaces, generalizing both the classical Abel-Jacobi theorem and its tropical analogue.

An Abel-Jacobi theorem for metrized complexes of Riemann surfaces

TL;DR

The paper proves an Abel-Jacobi theorem for metrized complexes of Riemann surfaces (mCRS), unifying the classical Abel-Jacobi theorem with its tropical analogue in hybrid geometric settings. By introducing the Jacobian and an Abel-Jacobi map , it provides a canonical isomorphism and shows that degree-zero divisors are principal precisely when they map to zero under . The result interpolates between and , recovers the classical Abel-Jacobi theorem for Riemann surfaces and the tropical theorem for metric graphs, and provides a robust framework for limit linear series in hybrid spaces. The methods combine Abel-Jacobi theory for components and graphs with a snake-lemma argument to derive the global statement on mCRS.

Abstract

Motivated by the recent surge of interest in the geometry of hybrid spaces, we prove an Abel-Jacobi theorem for a metrized complex of Riemann surfaces, generalizing both the classical Abel-Jacobi theorem and its tropical analogue.
Paper Structure (4 sections, 9 theorems, 79 equations, 3 figures)

This paper contains 4 sections, 9 theorems, 79 equations, 3 figures.

Table of Contents

  1. Divisors on metric graphs
  2. Divisors on metrized complexes of Riemann surfaces
  3. The Jacobian of a metrized complex of Riemann surfaces
  4. Abel--Jacobi theory for metrized complexes of Riemann surfaces

Key Result

Theorem 1

Let $\mathfrak{X}$ be a metrized complex of Riemann surfaces with underlying metric graph $\Gamma$ and with compact Riemann surfaces $X_v$ at the vertices $v$ of $\Gamma$. There is a canonical isomorphism of short exact sequences given by the following commutative diagram where all vertical maps are given by the respective Abel-Jacobi maps in degree zero. Furthermore, both rows are right-split, a

Figures (3)

  • Figure 1: Geometric realization of a metrized complex of Riemann surfaces of genus $3$.
  • Figure 2: Barycentirc subdivision of $\Delta^2$ under the assumption that one iteration is already sufficient. The zig-zag line from $\sigma(P)$ to $\sigma(Q)$ depicts $\sigma|_k$ whereas the smooth path depicts a possible choice of $\gamma_k$.
  • Figure 3: This picture shows the second case below. In the geometric realization $|\mathfrak{X}|$ of $\mathfrak{X}$, the line segment $L_j$ has Riemann surfaces $X_{v_j^i}$ with two marked points each in its interior.

Theorems & Definitions (30)

  • Theorem 1: Theorem \ref{['thm:abel_jacobi_for_mCRS']}
  • Corollary 2
  • Definition 1.1: Metric Graphs
  • Definition 1.2: Harmonic $1$-Forms
  • Theorem 1.3: Abel-Jacobi theorem for metric graphs
  • Definition 2.1
  • Example 2.2: Riemann surfaces as special cases of mCRS
  • Example 2.3: Metric graphs as special cases of mCRS
  • Proposition 3.1
  • Remark 3.2
  • ...and 20 more