## Abstract

A monomial self-map f on a complex toric variety is said to be κ-stable if the action induced on the 2κ-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 κ-stable. If f is replaced by an iterate one can find a k-stable model as soon as the dynamical degrees λ_{κ} of f satisfy λ2 _{κ}^{2} > λ_{κ-1}λ_{κ+1}. On the other hand, we give examples of monomial maps f, where this condition is not satisfied and where the degree sequences deg_{κ}(f^{n}) do not satisfy any linear recurrence. It follows that such an f is not κ-stable on any toric model with at worst quotient singularities.

Original language | English (US) |
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Pages (from-to) | 2127-2146 |

Number of pages | 20 |

Journal | Annales de l'Institut Fourier |

Volume | 64 |

Issue number | 5 |

DOIs | |

State | Published - 2014 |

## Keywords

- Algebraic stability
- Degree growth
- Monomial maps

## ASJC Scopus subject areas

- Algebra and Number Theory
- Geometry and Topology