AI 中文总结
该研究针对$\boldsymbol{\text{P}}^n$上$q>1$的$q$-极化自同态，证明其分歧对的对数典范性，在$n=2$时确定对数典范阈值下界并构造出对数Calabi-Yau对，完成了光滑射影曲面上Gongyo猜想的证明。
AI 中文摘要
设$f:\boldsymbol{\text{P}}^n\to\boldsymbol{\text{P}}^n$为$q>1$的$q$-极化自同态，$R_f$为其分歧除子，研究分歧对$(\boldsymbol{\text{P}}^n,R_f)$的奇点。证明对一般的$f$，该对是对数典范的；当$n=2$时，存在整数$s\geq1$使得对数典范阈值$\text{lct}(\boldsymbol{\text{P}}^2;R_{f^s})\text{≥}1/(q^s-1)$，迭代在一般情形下必要，该下界是最优的，特别地，$(\boldsymbol{\text{P}}^2,R_{f^s}/(q^s-1))$是对数Calabi-Yau对，完成了光滑射影曲面上Gongyo猜想的证明。
英文摘要
Let $f:\mathbf{P}^n\to\mathbf{P}^n$ be a $q$-polarized endomorphism, where $q>1$, and let $R_f$ be its ramification divisor. We study the singularities of the ramification pair $(\mathbf{P}^n,R_f)$. We show that, for a general $f$, the pair $(\mathbf{P}^n,R_f)$ is log canonical. When $n=2$, we prove that there exists an integer $s\geq1$ such that the log canonical threshold $\mathrm{lct}(\mathbf{P}^2;R_{f^s})\geq1/(q^s-1)$. The passage to an iterate is necessary in general, and the lower bound is optimal. In particular, $(\mathbf{P}^2,R_{f^s}/(q^s-1))$ is a log Calabi--Yau pair, completing the proof of Gongyo's conjecture for smooth projective surfaces.
Comments26 pages, comments are welcome!
