Lin, Jan-Li; Wulcan, Elizabeth
Stabilization of monomial maps in higher codimension  [ Stabilisation des applications monomiales en haute codimension ]
Annales de l'institut Fourier, Tome 64 (2014) no. 5 , p. 2127-2146
MR 3330933 | Zbl 06387333
doi : 10.5802/aif.2906
URL stable : http://www.numdam.org/item?id=AIF_2014__64_5_2127_0

Classification:  14M25,  37F10
Mots clés: stabilité algébrique, applications monomiales, croissance des degrés
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.
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.

Bibliographie

[1] Barrett, Wayne; Johnson, Charles R. Possible spectra of totally positive matrices, Linear Algebra Appl., 62 (1984), p. 231–233 Article  MR 761070 | Zbl 0551.15007

[2] Bedford, Eric; Kim, Kyounghee Linear recurrences in the degree sequences of monomial mappings, Ergodic Theory Dynam. Systems, 28 (2008) no. 5, p. 1369–1375 Article  MR 2449533 | Zbl 1161.37032

[3] Danilov, V. I. The geometry of toric varieties, Uspekhi Mat. Nauk, 33 (1978) no. 2(200), p. 85–134, 247 MR 495499 | Zbl 0425.14013

[4] Diller, J.; Favre, C. Dynamics of bimeromorphic maps of surfaces, Amer. J. Math., 123 (2001) no. 6, p. 1135–1169 Article  MR 1867314 | Zbl 1112.37308

[5] Dinh, Tien-Cuong; Sibony, Nessim Dynamics of regular birational maps in k , J. Funct. Anal., 222 (2005) no. 1, p. 202–216 Article  MR 2129771 | Zbl 1067.37055

[6] Dinh, Tien-Cuong; Sibony, Nessim Une borne supérieure pour l’entropie topologique d’une application rationnelle, Ann. of Math. (2), 161 (2005) no. 3, p. 1637–1644 Article  MR 2180409 | Zbl 1084.54013

[7] Dinh, Tien-Cuong; Sibony, Nessim Super-potentials of positive closed currents, intersection theory and dynamics, Acta Math., 203 (2009) no. 1, p. 1–82 Article  MR 2545825 | Zbl 1227.32024

[8] Favre, Charles Les applications monomiales en deux dimensions, Michigan Math. J., 51 (2003) no. 3, p. 467–475 Article  MR 2021001 | Zbl 1053.37021

[9] Favre, Charles; Jonsson, Mattias Dynamical compactifications of C 2 , Ann. of Math. (2), 173 (2011) no. 1, p. 211–248 Article  MR 2753603 | Zbl 1244.32012

[10] Favre, Charles; Wulcan, Elizabeth Degree growth of monomial maps and McMullen’s polytope algebra, Indiana Univ. Math. J., 61 (2012) no. 2, p. 493–524 Article  MR 3043585 | Zbl 1291.37058

[11] Fornaess, John Erik; Sibony, Nessim Complex dynamics in higher dimension. II, Modern methods in complex analysis (Princeton, NJ, 1992), Princeton Univ. Press, Princeton, NJ (Ann. of Math. Stud.) 137 (1995), p. 135–182 MR 1369137 | Zbl 0847.58059

[12] Fulton, William Introduction to toric varieties, Princeton University Press, Princeton, NJ, Annals of Mathematics Studies, 131 (1993), p. xii+157 (The William H. Roever Lectures in Geometry) MR 1234037 | Zbl 0813.14039

[13] Fulton, William Intersection theory, Springer-Verlag, Berlin, 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 (1998), p. xiv+470 MR 1644323 | Zbl 0541.14005

[14] Guedj, Vincent Ergodic properties of rational mappings with large topological degree, Ann. of Math. (2), 161 (2005) no. 3, p. 1589–1607 Article  MR 2179389 | Zbl 1088.37020

[15] Hasselblatt, Boris; Propp, James Degree-growth of monomial maps, Ergodic Theory Dynam. Systems, 27 (2007) no. 5, p. 1375–1397 Article  MR 2358970 | Zbl 1143.37032

[16] Huber, Birkett; Sturmfels, Bernd A polyhedral method for solving sparse polynomial systems, Math. Comp., 64 (1995) no. 212, p. 1541–1555 Article  MR 1297471 | Zbl 0849.65030

[17] Jonsson, Mattias; Wulcan, Elizabeth Stabilization of monomial maps, Michigan Math. J., 60 (2011) no. 3, p. 629–660 Article  MR 2861092 | Zbl 1247.37040

[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., 271 (2012) no. 1-2, p. 293–311 Article  MR 2917145 | Zbl 1247.32018

[20] Lin, Jan-Li Pulling back cohomology classes and dynamical degrees of monomial maps, Bull. Soc. Math. France, 140 (2012) no. 4, p. 533–549 (2013) Numdam | MR 3059849

[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, Springer-Verlag, Berlin, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 15 (1988), p. viii+212 (An introduction to the theory of toric varieties, Translated from the Japanese) MR 922894 | Zbl 0628.52002

[23] Peters, Chris A. M.; Steenbrink, Joseph H. M. Mixed Hodge structures, Springer-Verlag, Berlin, 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 (2008), p. xiv+470 MR 2393625 | Zbl 1138.14002

[24] Russakovskii, Alexander; Shiffman, Bernard Value distribution for sequences of rational mappings and complex dynamics, Indiana Univ. Math. J., 46 (1997) no. 3, p. 897–932 Article  MR 1488341 | Zbl 0901.58023

[25] Sibony, Nessim Dynamique des applications rationnelles de P k , Dynamique et géométrie complexes (Lyon, 1997), Soc. Math. France, Paris (Panor. Synthèses) 8 (1999), p. ix–x, xi–xii, 97–185 MR 1760844 | Zbl 1020.37026

[26] Stanley, Richard P. Enumerative combinatorics. Vol. I, Wadsworth & Brooks/Cole Advanced Books & Software, Monterey, CA, The Wadsworth & Brooks/Cole Mathematics Series (1986), p. xiv+306 (With a foreword by Gian-Carlo Rota) MR 847717 | Zbl 0608.05001