Pulling back cohomology classes and dynamical degrees of monomial maps
[Le pull-back de classes de cohomologie et les degrés dynamiques d'applications monomiales]
Lin, Jan-Li
Bulletin de la Société Mathématique de France, Tome 140 (2012), p. 533-549 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

We study the pullback maps on cohomology groups for equivariant rational maps (i.e., monomial maps) on toric varieties. Our method is based on the intersection theory on toric varieties. We use the method to determine the dynamical degrees of monomial maps and compute the degrees of the Cremona involution.

Publié le : 2012-01-01
DOI : https://doi.org/10.24033/bsmf.2635
Classification:  37F10,  14M25 
@article{BSMF_2012__140_4_533_0,
     author = {Lin, Jan-Li},
     title = {Pulling back cohomology classes and dynamical degrees of monomial maps},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     volume = {140},
     year = {2012},
     pages = {533-549},
     doi = {10.24033/bsmf.2635},
     mrnumber = {3059849},
     language = {en},
     url = {http://dml.mathdoc.fr/item/BSMF_2012__140_4_533_0}
}

Lin, Jan-Li. Pulling back cohomology classes and dynamical degrees of monomial maps. Bulletin de la Société Mathématique de France, Tome 140 (2012) pp. 533-549. doi : 10.24033/bsmf.2635. http://gdmltest.u-ga.fr/item/BSMF_2012__140_4_533_0/

[1] J.-C. Anglès D'Auriac, J.-M. Maillard & C. M. Viallet - « On the complexity of some birational transformations », J. Phys. A 39 (2006), p. 3641-3654. | MR 2220002 | Zbl 1086.14504

[2] E. Bedford & K. Kim - « On the degree growth of birational mappings in higher dimension », J. Geom. Anal. 14 (2004), p. 567-596. | MR 2111418 | Zbl 1067.37054

[3] E. Bedford & T. T. Truong - « Degree complexity of birational maps related to matrix inversion », Comm. Math. Phys. 298 (2010), p. 357-368. | MR 2669440 | Zbl 1205.15009

[4] V. I. Danilov - « The geometry of toric varieties », Russ. Math. Surv. 33 (1978), p. 97-154. | MR 495499 | Zbl 0425.14013

[5] J. Diller & C. Favre - « Dynamics of bimeromorphic maps of surfaces », Amer. J. Math. 123 (2001), p. 1135-1169. | MR 1867314 | Zbl 1112.37308

[6] T.-C. Dinh & V.-A. Nguyên - « Comparison of dynamical degrees for semi-conjugate meromorphic maps », Comment. Math. Helv. 86 (2011), p. 817-840. | MR 2851870 | Zbl 1279.32018

[7] T.-C. Dinh & N. Sibony - « Une borne supérieure pour l'entropie topologique d'une application rationnelle », Ann. of Math. 161 (2005), p. 1637-1644. | MR 2180409 | Zbl 1084.54013

[8] C. Favre - « Les applications monomiales en deux dimensions », Michigan Math. J. 51 (2003), p. 467-475. | MR 2021001 | Zbl 1053.37021

[9] C. Favre & M. Jonsson - « Dynamical compactifications of 𝐂 2 », Ann. of Math. 173 (2011), p. 211-248. | MR 2753603 | Zbl 1244.32012

[10] C. Favre & E. Wulcan - « Degree growth of monomial maps and mcmullen's polytope algebra », preprint arXiv:1011.2854, to appear in Indiana Univ. Math. J. | MR 3043585 | Zbl 1291.37058 | Zbl pre06170797

[11] W. Fulton - Introduction to toric varieties, Annals of Math. Studies, vol. 131, 1993. | MR 1234037 | Zbl 0813.14039

[12] -, Intersection theory, Springer, 1998.

[13] W. Fulton & B. Sturmfels - « Intersection theory on toric varieties », Topology 36 (1997), p. 335-353. | MR 1415592 | Zbl 0885.14025

[14] G. Gonzalez-Sprinberg & I. Pan - « On characteristic classes of determinantal Cremona transformations », Math. Ann. 335 (2006), p. 479-487. | MR 2221122 | Zbl 1097.14011

[15] V. Guedj - « Propriétés ergodiques des applications rationnelles », Panoramas & Synthèses 30 (2010), p. 97-202. | MR 2932434 | Zbl pre06114593

[16] B. Hasselblatt & J. Propp - « Degree-growth of monomial maps », Ergodic Theory Dynam. Systems 27 (2007), p. 1375-1397. | MR 2358970 | Zbl 1143.37032

[17] M. Jonsson & E. Wulcan - « Stabilization of monomial maps », Michigan Math. J. 60 (2011), p. 629-660. | MR 2861092 | Zbl 1247.37040

[18] J.-L. Lin - « Algebraic stability and degree growth of monomial maps », Math. Z. 271 (2012), p. 293-311. | MR 2917145 | Zbl 1247.32018

[19] V.-A. Nguyên - « Algebraic degrees for iterates of meromorphic self-maps of ℙ k », Publ. Mat. 50 (2006), p. 457-473. | MR 2273670 | Zbl 1112.37035

[20] K. Oguiso - « A remark on dynamical degrees of automorphisms of hyperkähler manifolds », Manuscripta Math. 130 (2009), p. 101-111. | MR 2533769 | Zbl 1172.14026

[21] A. Russakovskii & B. Shiffman - « Value distribution for sequences of rational mappings and complex dynamics », Indiana Univ. Math. J. 46 (1997), p. 897-932. | MR 1488341 | Zbl 0901.58023