Log Calabi--Yau structure for endomorphisms on Pⁿ

2026/08/03 by Yujie Luo, Sheng Meng
Mathematics · #math.AG #math.DS #msc:14E22 #msc:14B05 #msc:37F80

paper · pdf

26 pages, comments are welcome!

arxiv created 2026/08/03 · arxiv updated 2026/08/04

Abstract

Let f:Pⁿ\toPⁿ be a q-polarized endomorphism, where q>1, and let R_f be its ramification divisor. We study the singularities of the ramification pair (Pⁿ,R_f). We show that, for a general f, the pair (Pⁿ,R_f) is log canonical. When n=2, we prove that there exists an integer s≥1 such that the log canonical threshold lct(P²;R_f^s)≥1/(q^s-1). The passage to an iterate is necessary in general, and the lower bound is optimal. In particular, (P²,R_f^s/(q^s-1)) is a log Calabi--Yau pair, completing the proof of Gongyo's conjecture for smooth projective surfaces.

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