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

Jingyu Mu, Yiqian Shi, Tianyang Sun, Bin Xu

Published 2023-02-25Version 1

Let $n$ be a positive integer, $\gamma_1>-1,\cdots,\gamma_n>-1$, $D=\{z\in {\Bbb C}:|z|<1\}$, and $(a_{ij})_{n\times n}$ be the Cartan matrix of $\frak{su}(n+1)$. By using the Fuchsian equation of $(n+1)$th order around a singular source of ${\rm SU}(n+1)$ Toda system discovered by Lin-Wei-Ye ($\textit{Invent Math}$, $\textbf{190}$(1):169-207, 2012), we describe precisely 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*} in terms of some $(n+1)$ holomorphic functions satisfying the normalized condition. Moreover, we show that for each $1\leq i\leq n$, $0$ is the cone singularity with angle $2\pi(1+\gamma_i)$ for metric $e^{u_i}|{\rm d}z|^2$ on $D\backslash\{0\}$, whose restriction near $0$ could be characterized by some $(n-1)$ holomorphic functions non-vanishing at $0$.