Stabilization of monomial maps in higher codimension
Annales de l'Institut Fourier, Volume 64 (2014) no. 5, pp. 2127-2146.

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 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.

Une application monomiale f d’une variété torique complexe dans elle-même est dite k-stable si l’action induite sur le 2k-ème groupe de cohomologie est compatible avec l’itération. Nous démontrons que sous des conditions appropriées sur les valeurs propres de la matrice des exposants associés de f, il existe un modèle torique à singularités quotients pour laquelle f est k-stable. De plus, si l’on remplace f par une de ses itérés, l’existence d’un modèle torique k-stable pour f est garantie dès lors que les degrés dynamiques de f satisfont la condition λ k 2 >λ k-1 λ k+1 . Par ailleurs, nous donnons des exemples d’applications monomiales f pour lesquelles cette condition n’est pas satisfaite, et dont la suite de degrés deg k (f n ) ne satisfait aucune condition de récurrence linéaire. Il en résulte qu’une telle application f ne peut être k-stable pour aucune modèle torique à singularités quotients.

DOI: 10.5802/aif.2906
Classification: 14M25, 37F10
Keywords: Algebraic stability, monomial maps, degree growth
Mot clés : stabilité algébrique, applications monomiales, croissance des degrés
Lin, Jan-Li 1; Wulcan, Elizabeth 2

1 University of Notre Dame Department of Mathematics Notre Dame, IN 46556 (USA)
2 Chalmers University of Technology and the University of Gothenburg SE-412 96 Göteborg (Sweden)
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Lin, Jan-Li; Wulcan, Elizabeth. Stabilization of monomial maps in higher codimension. Annales de l'Institut Fourier, Volume 64 (2014) no. 5, pp. 2127-2146. doi : 10.5802/aif.2906. https://aif.centre-mersenne.org/articles/10.5802/aif.2906/

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