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 $$△u+e^u=4\pi \sum _{i=1}^N{\ell }_i{\delta }_{p_i}$$ on a flat torus $E=\mathbb{C}/\Lambda$, where $N,{\ell }_1,\dots ,{\ell }_N\in \mathbb{N}$, $p_i\in E$ 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 Lamé 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 Lamé 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 Lamé curves parametrizing logarithmic-free solutions is proposed.

50 pages, 5 figures