The simplicity of the first spectral radius of a meromorphic map

Abstract

Let $X$ be a compact Kahler manifold and let $f:X\rightarrow X$ be a dominant rational map which is 1-stable. Let $\lambda_1$ and $\lambda_2$ be the first and second dynamical degrees of $f$. If $\lambda_1^2>\lambda_2$, then we show that $\lambda_1$ is a simple eigenvalue of $f^*:H^{1,1}(X)\rightarrow H^{1,1}(X)$, and moreover the unique eigenvalue of modulus $>\sqrt{\lambda_2}$. A variant of the result, where we consider the first spectral radius in the case the map $f$ may not be 1-stable, is also given. An application is stated for bimeromorphic selfmaps of 3-folds. In the last section of the paper, we prove analogs of the above results in the algebraic setting, where $X$ is a projective manifold over an algebraic closed field of characteristic zero, and $f:X\rightarrow X$ is a rational map. Part of the section is devoted to defining dynamical degrees in the algebraic setting. We stress that here the dynamical degrees of rational maps can be defined over any algebraic closed field, not necessarily of characteristic zero.