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

2025/09/22 by W. L. Chang, De‐Qi Zhang, Chang, Wentao +1
Mathematics · #08A35 #14E30 #14J17 #32H50 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2509.17927

openalex publication_date 2025/09/22 · openalex created_date 2025/10/17 · openalex updated_date 2026/07/28

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 K_X+Δ∼_ℚ0. 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_Δ)/(q-1)) when (X,Δ) is an f-pair, i.e., K_X+Δ=f^*(K_X+Δ)+R_Δ with Δ, R_Δ being effective. In particular, we show: (1) K_X + (R_f)/(q-1) is ℚ-Cartier and numerically trivial when X is a ℚ-factorial (or of klt type) 3-fold; (2) (X, \fracR_fq-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.

Citations

• Baily--Borel compactifications of period images and the b-semiampleness conjecture
• Polarized endomorphisms of log Calabi-Yau pairs
• Log Calabi-Yau structure of projective threefolds admitting polarized endomorphisms
• Boundedness of elliptic Calabi-Yau threefolds
• On inversion of adjunction
• On connectedness of non-klt loci of singularities of pairs
• Structure of Fano fibrations of varieties admitting an int-amplified endomorphism
• Building blocks of amplified endomorphisms of normal projective varieties
• Building blocks of amplified endomorphisms of normal projective varieties
• Remarks on log Calabi--Yau structure of varieties admitting polarized endomorphism
• Anti-pluricanonical systems on Fano varieties
• Singularities of varieties admitting an endomorphism
• Inductive approach to effective b-semiampleness
• Abundance theorem for numerically trivial log canonical divisors of\n semi-log canonical pairs
• Minimal model theory for log surfaces
• Existence of minimal models for varieties of log general type
• Questions on self maps of algebraic varieties
• The Adjunction Conjecture and its applications

Related

• Log Calabi-Yau structure of projective threefolds admitting polarized endomorphisms
• Remarks on log Calabi--Yau structure of varieties admitting polarized endomorphism
• Polarized endomorphisms of log Calabi-Yau pairs
• Polarized endomorphisms of Fano varieties with complements
• Singularities of non-ℚ-Gorenstein varieties admitting a polarized endomorphism