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October 18, 2025Open Access

Log Calabi-Yau structure of algebaic varieties admitting a polarized endomorphism

Key Points

The study proves that under a polarized endomorphism, a Gorenstein terminal 3-fold exhibits Calabi-Yau type properties.
Key metrics indicate that when $X$ is $Q$-factorial, certain conditions hold for the numerically trivial property of $K_X + \frac{R_f}{q-1}$.
When examining $(X, \Delta)$ as an $f$-pair, necessary conditions for log Calabi-Yau structures are established for surfaces with specific Picard numbers.
The implications suggest that ongoing research into effective divisors can deepen understanding of complex projective varieties.

Abstract

Let X be a normal projective variety admitting a polarized endomorphism f, i. e. , f^*H qH for some ample divisor H and integer q>1. Then Broustet and Gongyo proposed the conjecture that X is of Calabi-Yau type (CY for short), i. e. , (X, Δ) is lc for some effective Q-divisor Δ and KX+Δₐ0. We prove the conjecture when X is a Gorenstein terminal 3-fold, extending the result of Sheng Meng for smooth threefolds. We then study the singularity type and CY property for (X, Δ+R_Δq-1) when (X, Δ) is an f-pair, i. e. , KX+Δ=f^* (KX+Δ) +R_Δ with Δ, R_Δ being effective. In particular, we show: (1) KX + Rfq-1 is Q-Cartier and numerically trivial when X is a Q-factorial (or of klt type) 3-fold; (2) (X, R₅q-1) is log Calabi-Yau when X is a surface with the Picard number ρ (X) >1 or f^-s (P) =P for some prime divisor P and s>0.

Log Calabi-Yau structure of algebaic varieties admitting a polarized endomorphism

Key Points

Abstract

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