Isoresidual curves

TL;DR

This paper studies the geometric and topological properties of isoresidual fibers in the moduli space of meromorphic differentials on the Riemann sphere, focusing on genus zero cases with two zeros.

Contribution

It provides a detailed description of the translation structure, invariants, and Euler characteristic of isoresidual fibers, including classification results for genus zero cases.

Findings

Generic isoresidual fibers are complex curves with a canonical translation structure.

The paper computes the Euler characteristic of these fibers using intersection theory.

It classifies connected components of isoresidual fibers in genus zero with multiple zeros.

Abstract

Given a partition $\mu$ of $-2$, the stratum $\mathcal{H}(\mu)$ parametrizes meromorphic differential one-forms on the Riemann sphere $\mathbb{CP}^{1}$ with~$n$ zeros and $p$ poles of orders prescribed by $\mu$. The isoresidual fibration is defined by assigning to each differential in $\mathcal{H}(\mu)$ its configuration of residues at the poles. In the case of differentials with $n=2$ zeros, generic isoresidual fibers are complex curves endowed with a canonical translation structure, which we describe extensively in this paper. Quantitative characteristics of the translation structure on isoresidual fiber curves, including the orders of the singularities and a period central charge encapsulating the linear dependence of periods on the underlying configuration of residues, provide rich discrete invariants for these fibers. We also determine the Euler characteristic of generic isoresidual fiber curves from intersection-theoretic computations, relying on the multi-scale compactification of strata of differentials. In particular, we describe a wall and chamber structure for the Euler characteristic of generic isoresidual fiber curves in terms of the partition $\mu$. Additionally, we classify the connected components of generic isoresidual fibers for strata in genus zero with an arbitrary number of zeros.