# Classifying solutions of ${\rm SU}(n+1)$ Toda system around a singular source

## TL;DR

This paper characterizes solutions to the ${
m SU}(n+1)$ Toda system around a singular source using holomorphic functions, providing a precise description of the solutions and the associated cone singularities.

## Contribution

It introduces a method to classify solutions of the ${
m SU}(n+1)$ Toda system near a singularity via $(n+1)$ holomorphic functions, extending previous differential equation approaches.

## Key findings

- Solutions are characterized by $(n+1)$ holomorphic functions.
- The singularity at 0 corresponds to a cone angle of $2\pi(1+\gamma_i)$.
- The metric near the singularity can be described by $(n-1)$ non-vanishing holomorphic functions.

## Abstract

Consider a positive integer $n$ and $\gamma_1>-1,\cdots,\gamma_n>-1$. Let $D=\{z\in {\Bbb C}:|z|<1\}$, and let $(a_{ij})_{n\times n}$ denote the Cartan matrix of $\frak{su}(n+1)$. Utilizing the ordinary differential equation of $(n+1)$th order around a singular source of ${\rm SU}(n+1)$ Toda system, as discovered by Lin-Wei-Ye ({\it Invent Math}, {\bf 190}(1):169-207, 2012), we precisely characterize a solution $(u_1,\cdots, u_n)$ to the ${\rm SU}(n+1)$ Toda system \begin{equation*} \begin{cases} \frac{\partial^2 u_i}{\partial z\partial \bar z}+\sum_{j=1}^n a_{ij} e^{u_j}&=\pi \gamma _i\delta _0\,\,{\rm on}\,\, D\\ \frac{\sqrt{-1}}{2}\,\int_{D\backslash \{0\}} e^{u_{i} }{\rm d}z\wedge {\rm d}\bar z &< \infty \end{cases} \quad \text{for all}\quad i=1,\cdots, n \end{equation*} using $(n+1)$ holomorphic functions that satisfy the normalized condition. Additionally, we demonstrate that for each $1\leq i\leq n$, $0$ represents the cone singularity with angle $2\pi(1+\gamma_i)$ for the metric $e^{u_i}|{\rm d}z|^2$ on $D\backslash\{0\}$, which can be locally characterized by $(n-1)$ non-vanishing holomorphic functions at $0$.