Stabilization of monomial maps in higher codimension

Jan-Li Lin[1]; Elizabeth Wulcan[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)

Annales de l’institut Fourier (2014)

  • Volume: 64, Issue: 5, page 2127-2146
  • ISSN: 0373-0956

Abstract

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A monomial self-map f on a complex toric variety is said to be k -stable if the action induced on the 2 k -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.

How to cite

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Lin, Jan-Li, and Wulcan, Elizabeth. "Stabilization of monomial maps in higher codimension." Annales de l’institut Fourier 64.5 (2014): 2127-2146. <http://eudml.org/doc/275441>.

@article{Lin2014,
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 $\lambda _k^2&gt;\lambda _\{k-1\}\lambda _\{k+1\}$. 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.},
affiliation = {University of Notre Dame Department of Mathematics Notre Dame, IN 46556 (USA); Chalmers University of Technology and the University of Gothenburg SE-412 96 Göteborg (Sweden)},
author = {Lin, Jan-Li, Wulcan, Elizabeth},
journal = {Annales de l’institut Fourier},
keywords = {Algebraic stability; monomial maps; degree growth; algebraic stability},
language = {eng},
number = {5},
pages = {2127-2146},
publisher = {Association des Annales de l’institut Fourier},
title = {Stabilization of monomial maps in higher codimension},
url = {http://eudml.org/doc/275441},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Lin, Jan-Li
AU - Wulcan, Elizabeth
TI - Stabilization of monomial maps in higher codimension
JO - Annales de l’institut Fourier
PY - 2014
PB - Association des Annales de l’institut Fourier
VL - 64
IS - 5
SP - 2127
EP - 2146
AB - 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 $\lambda _k^2&gt;\lambda _{k-1}\lambda _{k+1}$. 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.
LA - eng
KW - Algebraic stability; monomial maps; degree growth; algebraic stability
UR - http://eudml.org/doc/275441
ER -

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