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arXiv · 1206.4925

Stabilization of monomial maps in higher codimension

Jan-Li Lin·Elizabeth Wulcan
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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 $λ_k$ of $f$ satisfy $λ_k^2>λ_{k-1}λ_{k+1}$. On the other hand, we give examples of monomial maps $f$, where this condition is not satisfied and where the degree sequences $°_k(f^n)$ 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.

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Jan-Li Lin, Elizabeth Wulcan. 2013-04-03. Stabilization of monomial maps in higher codimension. https://arxiv.org/abs/1206.4925

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