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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.22175 |
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| _version_ | 1866908990801182720 |
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| author | Wang, Chin-Lung |
| author_facet | Wang, Chin-Lung |
| contents | We explain how algebraic geometry comes into play in the study of non-linear mean field (singular Liouville) equations $$ \triangle u + e^u = 4π\sum_{i = 1}^N \ell_i δ_{p_i} $$ on a flat torus $E = \Bbb C/Λ$, where $N, \ell_1, \ldots, \ell_N \in \Bbb N$, $p_i \in E$ are distinct points, and $δ_{p_i}$ 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 $Z_n(σ, τ)$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_22175 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Algebraic methods in periodic singular Liouville equations Wang, Chin-Lung Algebraic Geometry Mathematical Physics Classical Analysis and ODEs 14H70, 35J61, 34M35 We explain how algebraic geometry comes into play in the study of non-linear mean field (singular Liouville) equations $$ \triangle u + e^u = 4π\sum_{i = 1}^N \ell_i δ_{p_i} $$ on a flat torus $E = \Bbb C/Λ$, where $N, \ell_1, \ldots, \ell_N \in \Bbb N$, $p_i \in E$ are distinct points, and $δ_{p_i}$ 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 $Z_n(σ, τ)$ 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. |
| title | Algebraic methods in periodic singular Liouville equations |
| topic | Algebraic Geometry Mathematical Physics Classical Analysis and ODEs 14H70, 35J61, 34M35 |
| url | https://arxiv.org/abs/2604.22175 |