Algebraic methods in periodic singular Liouville equations

Abstract

We explain how algebraic geometry comes into play in the study of non-linear mean field (singular Liouville) equations $$ \triangle u + e^u = 4\pi \sum_{i = 1}^N \ell_i \delta_{p_i} $$ on a flat torus $E = \Bbb C/\Lambda$, where $N, \ell_1, \ldots, \ell_N \in \Bbb N$, $p_i \in E$ are distinct points, and $\delta_{p_i}$ is the Dirac measure at $p_i$. The case with one singular source ($N = 1$) had been studied extensively in recent years. We start with a survey of this case with emphasizes on the constructions of Lam\'e curves $\overline X_n$ and pre-modular forms $Z_n(\sigma, \tau)$ which encodes the structure of solutions of the PDE. We then discuss extensions to the case of general $N$. The basic tool is the monodromy theory for generalized Lam\'e equations. Two aspects are discussed: (1) For $\ell := \sum_{i = 1}^N \ell_i$ being odd, an exact counting formula of \emph{algebraic degree} is proved. (2) For $\ell$ being even, the existence of generalized Lam\'e curves parametrizing logarithmic-free solutions is proposed.