Stabilization of monomial maps in higher codimension

TL;DR

This paper investigates conditions under which monomial maps on complex toric varieties can be stabilized in higher codimension, providing criteria based on eigenvalues and dynamical degrees, and illustrating cases where stabilization is impossible.

Contribution

It establishes new conditions for k-stability of monomial maps on toric varieties and demonstrates when such stability can or cannot be achieved.

Findings

k-stability can be achieved under eigenvalue conditions

Stability depends on the inequality involving dynamical degrees

Examples show instability when conditions are not met

Abstract

A monomial self-map $f$ on a complex toric variety is said to be $k$-stable if the action induced on the $2k$-cohomology is compatible with iteration. We show that under suitable conditions on the eigenvalues of the matrix of exponents of $f$, we can find a toric model with at worst quotient singularities where $f$ is $k$-stable. If $f$ is replaced by an iterate one can find a $k$-stable model as soon as the dynamical degrees $\lambda_k$ of $f$ satisfy $\lambda_k^2>\lambda_{k-1}\lambda_{k+1}$. On the other hand, we give examples of monomial maps $f$, where this condition is not satisfied and where the degree sequences $\deg_k(f^n)$ do not satisfy any linear recurrence. It follows that such an $f$ is not $k$-stable on any toric model with at worst quotient singularities.