Algebraic entropy for smooth projective varieties

We show that the spectral radius for the action of a self map $f$ of a smooth projective variety (over an arbitrary base field) on its $\ell$‑adic cohomology is achieved on the $f^\ast$ stable sub-algebra generated by any ample class. This generalizes a result of Esnault–Srinivas who had obtained an analogous result for automorphisms of surfaces. Over $\mathbb{C}$ we also show that this sub-algebra is naturally an irreducible representation of a Looijenga–Lunts–Verbitsky type Lie algebra acting on the cohomology of a smooth projective variety.

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Received 25 September 2020

Accepted 23 February 2021

Published 30 November 2022