Mathematical Research Letters

Volume 30 (2023)

Number 2

Rigidity of rationally connected smooth projective varieties from dynamical viewpoints

Pages: 589 – 610



Sheng Meng (School of Mathematical Sciences, Key Laboratory of MEA (Ministry of Education) & Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai, China; and School of Mathematics, Korea Institute For Advanced Study, Seoul, South Korea)

Guolei Zhong (Department of Mathematics, National University of Singapore)


Let $X$ be a rationally connected smooth projective variety of dimension $n$. We show that $X$ is a toric variety if and only if $X$ admits an int-amplified endomorphism with totally invariant ramification divisor. We also show that $X \cong (\mathbb{P}^1)^{\times n}$ if and only if $X$ admits a surjective endomorphism $f$ such that the eigenvalues of $f^\ast \vert_{\mathrm{N}^1(X)}$ (without counting multiplicities) are $n$ distinct real numbers greater than $1$.

Received 26 February 2021

Received revised 23 October 2021

Accepted 8 November 2021

Published 13 September 2023