An inequality on polarized endomorphisms

TL;DR

This paper proves an inequality relating the norm and degree of polarized endomorphisms on smooth projective varieties, assuming standard conjectures, extending a known Kähler case result.

Contribution

It establishes a new inequality for polarized endomorphisms under standard conjectures, confirming a conjecture and extending Serre's Kähler result to algebraic varieties.

Findings

The inequality holds assuming standard conjectures.

It confirms the conjecture for polarized endomorphisms.

Extends Serre's Kähler result to algebraic varieties.

Abstract

We show that assuming the standard conjectures, for any smooth projective variety $X$ of dimension $n$ over an algebraically closed field, there is a constant $C>0$ such that for any positive rational number $r$ and for any polarized endomorphism $f$ of $X$, we have \[ \| G_r \circ f \| \le C \, \mathrm{deg}(G_r \circ f), \] where $G_r$ is a correspondence of $X$ so that for each $0\le i\le 2n$ its pullback action on the $i$-th Weil cohomology group is the multiplication-by-$r^i$ map. This inequality has been conjectured by the authors to hold in a more general setting, which - in the special case of polarized endomorphisms - confirms the validity of the analog of a well known result by Serre in the K\"ahler setting.