Residue polytopes

TL;DR

This paper introduces residue polytopes derived from level graphs, demonstrating their face structure and compatibility, with applications to the limits of Abelian differentials on Riemann surfaces.

Contribution

It defines residue polytopes for level graphs, proves their face relations, and connects these structures to the geometry of Riemann surface degenerations.

Findings

Residue polytopes correspond to level graphs and their face relations.

Residue polytopes form all faces of the polytope associated with the trivial partition.

Applications include describing limits of Abelian differentials on degenerating Riemann surfaces.

Abstract

A level graph is the data of a pair $(G,\pi)$ consisting of a finite graph $G$ and an ordered partition $\pi$ on the set of vertices of $G$. To each level graph on $n$ vertices we associate a polytope in $\mathbb R^n$ called its residue polytope. We show that residue polytopes are compatible with each other in the sense that if $\pi'$ is a coarsening of $\pi$, then the polytope associated to $(G,\pi)$ is a face of the one associated to $(G,\pi')$. Moreover, they form all the faces of the residue polytope of $G$, defined as the polytope associated to the level graph with the trivial ordered partition. The results are used in a companion work to describe limits of spaces of Abelian differentials on families of Riemann surfaces approaching a stable Riemann surface on the boundary of the moduli space.