Stabilization of monomial maps in higher codimension

Lin, Jan-Li; Wulcan, Elizabeth

doi:10.5802/aif.2906

Stabilization of monomial maps in higher codimension
[Stabilisation des applications monomiales en haute codimension]

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)

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

Résumé
Abstract

Une application monomiale $f$ d’une variété torique complexe dans elle-même est dite $k$ -stable si l’action induite sur le $2 k$ -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.

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.

MR Zbl

DOI : 10.5802/aif.2906

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

Affiliations des auteurs :

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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     author = {Lin, Jan-Li and Wulcan, Elizabeth},
     title = {Stabilization of monomial maps in higher codimension},
     journal = {Annales de l'Institut Fourier},
     pages = {2127--2146},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {64},
     number = {5},
     year = {2014},
     doi = {10.5802/aif.2906},
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Lin, Jan-Li; Wulcan, Elizabeth. Stabilization of monomial maps in higher codimension. Annales de l'Institut Fourier, Tome 64 (2014) no. 5, pp. 2127-2146. doi : 10.5802/aif.2906. https://aif.centre-mersenne.org/articles/10.5802/aif.2906/

Bibliographie
Cité par

[1] Barrett, Wayne; Johnson, Charles R. Possible spectra of totally positive matrices, Linear Algebra Appl., Volume 62 (1984), pp. 231-233 | DOI | MR | Zbl

[2] Bedford, Eric; Kim, Kyounghee Linear recurrences in the degree sequences of monomial mappings, Ergodic Theory Dynam. Systems, Volume 28 (2008) no. 5, pp. 1369-1375 | DOI | MR | Zbl

[3] Danilov, V. I. The geometry of toric varieties, Uspekhi Mat. Nauk, Volume 33 (1978) no. 2(200), p. 85-134, 247 | MR | Zbl

[4] Diller, J.; Favre, C. Dynamics of bimeromorphic maps of surfaces, Amer. J. Math., Volume 123 (2001) no. 6, pp. 1135-1169 | DOI | MR | Zbl

[5] Dinh, Tien-Cuong; Sibony, Nessim Dynamics of regular birational maps in $ℙ^{k}$ , J. Funct. Anal., Volume 222 (2005) no. 1, pp. 202-216 | DOI | MR | Zbl

[6] Dinh, Tien-Cuong; Sibony, Nessim Une borne supérieure pour l’entropie topologique d’une application rationnelle, Ann. of Math. (2), Volume 161 (2005) no. 3, pp. 1637-1644 | DOI | MR | Zbl

[7] Dinh, Tien-Cuong; Sibony, Nessim Super-potentials of positive closed currents, intersection theory and dynamics, Acta Math., Volume 203 (2009) no. 1, pp. 1-82 | DOI | MR | Zbl

[8] Favre, Charles Les applications monomiales en deux dimensions, Michigan Math. J., Volume 51 (2003) no. 3, pp. 467-475 | DOI | MR | Zbl

[9] Favre, Charles; Jonsson, Mattias Dynamical compactifications of $C^{2}$ , Ann. of Math. (2), Volume 173 (2011) no. 1, pp. 211-248 | DOI | MR | Zbl

[10] Favre, Charles; Wulcan, Elizabeth Degree growth of monomial maps and McMullen’s polytope algebra, Indiana Univ. Math. J., Volume 61 (2012) no. 2, pp. 493-524 | DOI | MR | Zbl

[11] Fornaess, John Erik; Sibony, Nessim Complex dynamics in higher dimension. II, Modern methods in complex analysis (Princeton, NJ, 1992) (Ann. of Math. Stud.), Volume 137, Princeton Univ. Press, Princeton, NJ, 1995, pp. 135-182 | MR | Zbl

[12] Fulton, William Introduction to toric varieties, Annals of Mathematics Studies, 131, Princeton University Press, Princeton, NJ, 1993, pp. xii+157 (The William H. Roever Lectures in Geometry) | MR | Zbl

[13] Fulton, William Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 2, Springer-Verlag, Berlin, 1998, pp. xiv+470 | MR | Zbl

[14] Guedj, Vincent Ergodic properties of rational mappings with large topological degree, Ann. of Math. (2), Volume 161 (2005) no. 3, pp. 1589-1607 | DOI | MR | Zbl

[15] Hasselblatt, Boris; Propp, James Degree-growth of monomial maps, Ergodic Theory Dynam. Systems, Volume 27 (2007) no. 5, pp. 1375-1397 | DOI | MR | Zbl

[16] Huber, Birkett; Sturmfels, Bernd A polyhedral method for solving sparse polynomial systems, Math. Comp., Volume 64 (1995) no. 212, pp. 1541-1555 | DOI | MR | Zbl

[17] Jonsson, Mattias; Wulcan, Elizabeth Stabilization of monomial maps, Michigan Math. J., Volume 60 (2011) no. 3, pp. 629-660 | DOI | MR | Zbl

[18] Lin, Jan-Li On Degree Growth and Stabilization of Three Dimensional Monomial Maps Jan-Li Lin (Michigan Math. J., to appear)

[19] Lin, Jan-Li Algebraic stability and degree growth of monomial maps, Math. Z., Volume 271 (2012) no. 1-2, pp. 293-311 | DOI | MR | Zbl

[20] Lin, Jan-Li Pulling back cohomology classes and dynamical degrees of monomial maps, Bull. Soc. Math. France, Volume 140 (2012) no. 4, p. 533-549 (2013) | Numdam | MR

[21] Mustaţă, M. Lecture notes on toric varieties (Available on the author’s webpage: www.math.lsa.umich.edu/~mmustata)

[22] Oda, Tadao Convex bodies and algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 15, Springer-Verlag, Berlin, 1988, pp. viii+212 (An introduction to the theory of toric varieties, Translated from the Japanese) | MR | Zbl

[23] Peters, Chris A. M.; Steenbrink, Joseph H. M. Mixed Hodge structures, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], 52, Springer-Verlag, Berlin, 2008, pp. xiv+470 | MR | Zbl

[24] Russakovskii, Alexander; Shiffman, Bernard Value distribution for sequences of rational mappings and complex dynamics, Indiana Univ. Math. J., Volume 46 (1997) no. 3, pp. 897-932 | DOI | MR | Zbl

[25] Sibony, Nessim Dynamique des applications rationnelles de $P^{k}$ , Dynamique et géométrie complexes (Lyon, 1997) (Panor. Synthèses), Volume 8, Soc. Math. France, Paris, 1999, p. ix-x, xi–xii, 97–185 | MR | Zbl

[26] Stanley, Richard P. Enumerative combinatorics. Vol. I, The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Monterey, CA, 1986, pp. xiv+306 (With a foreword by Gian-Carlo Rota) | MR | Zbl

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