# 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

## Access Full Article

top## Abstract

top## How to cite

topLin, 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 $\lambda _k^2>\lambda _\{k-1\}\lambda _\{k+1\}$. 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 $\lambda _k^2>\lambda _{k-1}\lambda _{k+1}$. 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 -

## References

top- Wayne Barrett, Charles R. Johnson, Possible spectra of totally positive matrices, Linear Algebra Appl. 62 (1984), 231-233 Zbl0551.15007MR761070
- Eric Bedford, Kyounghee Kim, Linear recurrences in the degree sequences of monomial mappings, Ergodic Theory Dynam. Systems 28 (2008), 1369-1375 Zbl1161.37032MR2449533
- V. I. Danilov, The geometry of toric varieties, Uspekhi Mat. Nauk 33 (1978), 85-134, 247 Zbl0425.14013MR495499
- J. Diller, C. Favre, Dynamics of bimeromorphic maps of surfaces, Amer. J. Math. 123 (2001), 1135-1169 Zbl1112.37308MR1867314
- Tien-Cuong Dinh, Nessim Sibony, Dynamics of regular birational maps in ${\mathbb{P}}^{k}$, J. Funct. Anal. 222 (2005), 202-216 Zbl1067.37055MR2129771
- Tien-Cuong Dinh, Nessim Sibony, Une borne supérieure pour l’entropie topologique d’une application rationnelle, Ann. of Math. (2) 161 (2005), 1637-1644 Zbl1084.54013MR2180409
- Tien-Cuong Dinh, Nessim Sibony, Super-potentials of positive closed currents, intersection theory and dynamics, Acta Math. 203 (2009), 1-82 Zbl1227.32024MR2545825
- Charles Favre, Les applications monomiales en deux dimensions, Michigan Math. J. 51 (2003), 467-475 Zbl1053.37021MR2021001
- Charles Favre, Mattias Jonsson, Dynamical compactifications of ${\mathbf{C}}^{2}$, Ann. of Math. (2) 173 (2011), 211-248 Zbl1244.32012MR2753603
- Charles Favre, Elizabeth Wulcan, Degree growth of monomial maps and McMullen’s polytope algebra, Indiana Univ. Math. J. 61 (2012), 493-524 Zbl1291.37058MR3043585
- John Erik Fornaess, Nessim Sibony, Complex dynamics in higher dimension. II, Modern methods in complex analysis (Princeton, NJ, 1992) 137 (1995), 135-182, Princeton Univ. Press, Princeton, NJ Zbl0847.58059MR1369137
- William Fulton, Introduction to toric varieties, 131 (1993), Princeton University Press, Princeton, NJ Zbl0813.14039MR1234037
- William Fulton, Intersection theory, 2 (1998), Springer-Verlag, Berlin Zbl0541.14005MR1644323
- Vincent Guedj, Ergodic properties of rational mappings with large topological degree, Ann. of Math. (2) 161 (2005), 1589-1607 Zbl1088.37020MR2179389
- Boris Hasselblatt, James Propp, Degree-growth of monomial maps, Ergodic Theory Dynam. Systems 27 (2007), 1375-1397 Zbl1143.37032MR2358970
- Birkett Huber, Bernd Sturmfels, A polyhedral method for solving sparse polynomial systems, Math. Comp. 64 (1995), 1541-1555 Zbl0849.65030MR1297471
- Mattias Jonsson, Elizabeth Wulcan, Stabilization of monomial maps, Michigan Math. J. 60 (2011), 629-660 Zbl1247.37040MR2861092
- Jan-Li Lin, On Degree Growth and Stabilization of Three Dimensional Monomial Maps Jan-Li Lin Zbl1300.37033
- Jan-Li Lin, Algebraic stability and degree growth of monomial maps, Math. Z. 271 (2012), 293-311 Zbl1247.32018MR2917145
- Jan-Li Lin, Pulling back cohomology classes and dynamical degrees of monomial maps, Bull. Soc. Math. France 140 (2012), 533-549 (2013) Zbl1333.37031MR3059849
- M. Mustaţă, Lecture notes on toric varieties Zbl1092.14064
- Tadao Oda, Convex bodies and algebraic geometry, 15 (1988), Springer-Verlag, Berlin Zbl0628.52002MR922894
- Chris A. M. Peters, Joseph H. M. Steenbrink, Mixed Hodge structures, 52 (2008), Springer-Verlag, Berlin Zbl1138.14002MR2393625
- Alexander Russakovskii, Bernard Shiffman, Value distribution for sequences of rational mappings and complex dynamics, Indiana Univ. Math. J. 46 (1997), 897-932 Zbl0901.58023MR1488341
- Nessim Sibony, Dynamique des applications rationnelles de ${\mathbf{P}}^{k}$, Dynamique et géométrie complexes (Lyon, 1997) 8 (1999), ix-x, xi–xii, 97–185, Soc. Math. France, Paris Zbl1020.37026MR1760844
- Richard P. Stanley, Enumerative combinatorics. Vol. I, (1986), Wadsworth & Brooks/Cole Advanced Books & Software, Monterey, CA Zbl0608.05001MR847717

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.