Algebraic entropy for smooth projective varieties

Abstract

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.

Added a new section on relationship with LLV Lie algebra and a new Corollary 3.8. Final version to appear in Math. Res. Lett

References in corpus (3)

• The Elementary Theory of the Frobenius Automorphisms
• Some aspects of explicit birational geometry inspired by complex dynamics
• Relations between dynamical degrees, Weil's Riemann hypothesis and the standard conjectures

Cited by in corpus (1)

• The Hrushovski-Lang-Weil estimates