Stabilization of monomial maps in higher codimension

Lin, Jan-Li; Wulcan, Elizabeth

Annales de l'Institut Fourier, Tome 64 (2014) no. 5, pp. 2127-2146.

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 ${\lambda }_k^2>{\lambda }_{k-1}{\lambda }_{k+1}$. Par ailleurs, nous donnons des exemples d’applications monomiales $f$ pour lesquelles cette condition n’est pas satisfaite, et dont la suite de degrés ${\mathrm{deg}}_k\left(f^n\right)$ 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.

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 ${deg}_k\left(f^n\right)$ 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.

Classification : 14M25, 37F10
Keywords: Algebraic stability, monomial maps, degree growth
Mot clés : stabilité algébrique, applications monomiales, croissance des degrés

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