Regularizations of pseudo-automorphisms with positive algebraic entropy

Alexandra Kuznetsova

doi:10.3934/jmd.2023006

Article Contents

2023, Volume 19: 237-268. Doi: 10.3934/jmd.2023006

This volume Previous Article On the Lagrange and Markov dynamical spectra for geodesic flows on surfaces with negative curvature Next Article The limit set of non-orientable mapping class groups

Regularizations of pseudo-automorphisms with positive algebraic entropy

Alexandra Kuznetsova^1,2,3,

1.
National Research University Higher School of Economics, 6 Usacheva St., Moscow 119048, Russia
2.
École Polytechnique, CMLS, Route de Saclay, 91128 Palaiseau, France
3.
Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkina St., Moscow 119991, Russia

Received: April 2022

Revised: October 21, 2022

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

We study birational automorphisms of threefolds which have positive algebraic entropy. We identify some conditions which imply that such an automorphism is non-regularizable. We show that this criterion applies in the example of a birational automorphism of $ \mathbb{P}^3 $ of positive algebraic entropy constructed by Blanc, thus showing that for a general choice of parameters it is non-regularizable. Additionally, we establish a criterion which proves that the automorphism in this example does not preserve a structure of a fibration over a surface.

Keywords:

Birational automorphisms,
dynamical degrees.

Mathematics Subject Classification: Primary: 14J50; Secondary: 14E07, 32H50, 37F10.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	E. Bedford, S. Cantat and K. Kim, Pseudo-automorphisms with no invariant foliation, J. Mod. Dyn., 8 (2014), 221-250. doi: 10.3934/jmd.2014.8.221.
[2]	E. Bedford and K. Kim, Dynamics of rational surface automorphisms: Linear fractional recurrences, J. Geom. Anal., 19 (2009), 553-583. doi: 10.1007/s12220-009-9077-8.
[3]	E. Bedford and K. Kim, Dynamics of (pseudo) automorphisms of 3-space: Periodicity versus positive entropy, Publ. Mat., 58 (2014), 65-119. doi: 10.5565/PUBLMAT_58114_04.
[4]	M. P. Bellon and C.-M. Viallet, Algebraic entropy, Comm. Math. Phys., 204 (1999), 425-437. doi: 10.1007/s002200050652.
[5]	J. Blanc, On the inertia group of elliptic curves in the Cremona group of the plane, Michigan Math. J., 56 (2008), 315-330. doi: 10.1307/mmj/1224783516.
[6]	J. Blanc, Dynamical degrees of (pseudo)-automorphisms fixing cubic hypersurfaces, Indiana Univ. Math. J., 62 (2013), 1143-1164. doi: 10.1512/iumj.2013.62.5040.
[7]	J. Blanc and S. Cantat, Dynamical degrees of birational transformations of projective surfaces, J. Amer. Math. Soc., 29 (2016), 415-471. doi: 10.1090/jams831.
[8]	S. Boucksom, J.-P. Demailly, M. Păun and T. Peternell, The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension, J. Algebraic Geom., 22 (2013), 201-248. doi: 10.1090/S1056-3911-2012-00574-8.
[9]	S. Cantat, J. Déserti and J. Xie, Three chapters on Cremona groups, Indiana Univ. Math. J., 70 (2021), 2011-2064. doi: 10.1512/iumj.2021.70.9153.
[10]	N.-B. Dang and C. Favre, Spectral interpretations of dynamical degrees and applications, Ann. of Math., 194 (2021), 299-359. doi: 10.4007/annals.2021.194.1.5.
[11]	J. Diller and C. Favre, Dynamics of bimeromorphic maps of surfaces, Amer. J. Math., 123 (2001), 1135-1169. doi: 10.1353/ajm.2001.0038.
[12]	T.-C. Dinh and V.-A. Nguyên, Comparison of dynamical degrees for semi-conjugate meromorphic maps, Comment. Math. Helv., 86 (2011), 817-840. doi: 10.4171/CMH/241.
[13]	T.-C. Dinh and N. Sibony, Une borne supérieure pour l'entropie topologique d'une application rationnelle, Ann. of Math. (2), 161 (2005), 1637-1644. doi: 10.4007/annals.2005.161.1637.
[14]	I. Dolgachev and D. Ortland, Point sets in projective spaces and theta functions, Astérisque, 165 (1988), 210 pp.
[15]	W. Fulton, Intersection Theory, second edition, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 2, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-1-4612-1700-8.
[16]	C. D. Hacon and J. McKernan, Flips and flops, in Proceedings of the International Congress of Mathematicians. Volume II, 513–539, Hindustan Book Agency, New Delhi, 2010.
[17]	R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, no. 52, Springer-Verlag, New York-Heidelberg, 1977.
[18]	J. Kollár, Lectures on Resolution of Singularities, Annals of Mathematics Studies, vol. 166, Princeton University Press, Princeton, NJ, 2007.
[19]	J. Kollár and S. Mori, Birational Geometry of Algebraic Varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998. doi: 10.1017/CBO9780511662560.
[20]	J. Lesieutre, Some constraints of positive entropy automorphisms of smooth threefolds, Ann. Sci. Éc. Norm. Supér. (4), 51 (2018), 1507–1547. doi: 10.24033/asens.2380.
[21]	F. Lo Bianco, On the cohomological action of automorphisms of compact Kähler threefolds, Bull. Soc. Math. France, 147 (2019), 469-514.
[22]	K. Oguiso and T. T. Truong, Explicit examples of rational and Calabi-Yau threefolds with primitive automorphisms of positive entropy, J. Math. Sci. Univ. Tokyo, 22 (2015), 361-385.
[23]	Yu. Prokhorov and C. Shramov, Jordan property for groups of birational selfmaps, Compos. Math., 150 (2014), 2054-2072. doi: 10.1112/S0010437X14007581.
[24]	T. T. Truong, The simplicity of the first spectral radius of a meromorphic map, Michigan Math. J., 63 (2014), 623-633. doi: 10.1307/mmj/1409932635.
[25]	T. T. Truong, Relative dynamical degrees of correspondences over a field of arbitrary characteristic, J. Reine Angew. Math., 758 (2020), 139-182. doi: 10.1515/crelle-2017-0052.
[26]	A. Weil, On algebraic groups of transformations, Amer. J. Math., 77 (1955), 355-391. doi: 10.2307/2372535.