Salem numbers in dynamics of Kähler threefolds and complex tori

Abstract

Let $X$ be a compact Kähler manifold of dimension $k\le 4$ and $f:X\to X$ a pseudo-automorphism. If the first dynamical degree ${\lambda }_1(f)$ is a Salem number, we show that either ${\lambda }_1(f)={\lambda }_{k-1}(f)$ or ${\lambda }_1(f{)}^2={\lambda }_{k-2}(f)$. In particular, if $\mbox{dim}(X)=3$ then ${\lambda }_1(f)={\lambda }_2(f)$. We use this to show that if $X$ is a complex 3-torus and $f$ is an automorphism of $X$ with ${\lambda }_1(f)>1$, then $f$ has a non-trivial equivariant holomorphic fibration if and only if ${\lambda }_1(f)$ is a Salem number. If $X$ is a complex 3-torus having an automorphism $f$ with ${\lambda }_1(f)={\lambda }_2(f)>1$ 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.

25 pages. Accepted in Math. Zeit

References in corpus (1)

• The Voronoi conjecture for parallelohedra with simply connected $\delta$-surface

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• Dynamical degrees of automorphisms on abelian varieties
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