Gongyo's log-canonicity conjecture for ramification divisors

Let XX be a smooth projective variety and let f ⁣:XXf\colon X\to X be a qq-polarized endomorphism, meaning that fHqHf^*H\sim qH for some ample divisor HH and integer q>1q>1. Let RfR_f be the ramification divisor defined by

KX=fKX+Rf.K_X=f^*K_X+R_f.

Gongyo's conjecture. After replacing ff by an iterate, the pair

(X,Rfq1)\left(X,\frac{R_f}{q-1}\right)

is log canonical. The conjecture is known in dimension two except possibly when XP2X\cong\mathbb{P}^2 and the sum of all f1f^{-1}-periodic prime divisors is zero; its general status is open.

Sources & referencesView supporting material

Primary source

Wentao Chang and De-Qi Zhang, “Log Calabi-Yau structure of algebaic varieties admitting a polarized endomorphism”, arXiv:2509.17927 (2025).

Additional references

3 papers in this index state this conjecture (2022–2025). The statement above is taken from the most recent of them; the others are arXiv:2311.16369, arXiv:2204.11244.

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