AI 中文总结
本文研究有限群在复环面上的FPF作用，证明每个FPF群有唯一不可约有理FPF表示，并给出非自由点、奇异点及orbifold欧拉示性数的闭式公式。
AI 中文摘要
设$G$为一个有限群，其允许一个有理无不动点（FPF）表示$V$，且$V$中存在一个$G$不变的满秩格$L\subset V$。我们研究$G$在复环面$V_{\mathbb{C}}/L$上的诱导作用以及商orbifold $V_{\mathbb{C}}/(G\ltimes L)$。我们证明了每个FPF群都允许一个唯一的不可约有理FPF表示。我们推导出三个核心几何不变量的闭式公式：环面作用的非自由点数、商orbifold的奇异点数及其orbifold欧拉示性数。
英文摘要
Let $G$ be a finite group admitting a rational fixed-point-free (FPF) representation $V$, with a $G$-invariant full-rank lattice $L\subset V$. We study the induced $G$-action on the complex torus $V_{\mathbb{C}}/L$ and the quotient orbifold $V_{\mathbb{C}}/(G\ltimes L)$. We prove that every FPF group admits a unique irreducible rational FPF representation. We derive closed-form formulas for three core geometric invariants: the number of non-free points of the torus action, the number of singular points of the quotient orbifold, and its orbifold Euler characteristic.
Comments19 pages
