Salem numbers in dynamics of Kähler threefolds and complex tori

2013/09/19 by Keiji Oguiso, Oguiso, Keiji, Tuyen Trung Truong +1 · 1 citation
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1309.4851

openalex publication_date 2013/09/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let X be a compact Kähler manifold of dimension k≤ 4 and f:X→ X a pseudo-automorphism. If the first dynamical degree λ₁(f) is a Salem number, we show that either λ₁(f)=λ_k-1(f) or λ₁(f)²=λ_k-2(f). In particular, if dim(X)=3 then λ₁(f)=λ₂(f). We use this to show that if X is a complex 3-torus and f is an automorphism of X with λ₁(f)>1, then f has a non-trivial equivariant holomorphic fibration if and only if λ₁(f) is a Salem number. If X is a complex 3-torus having an automorphism f with λ₁(f)=λ₂(f)>1 but is not a Salem number, then the Picard number of X must be 0,3 or 9, and all these cases can be realized.

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