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Transactions of the American Mathematical Society

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Some dynamical properties of pseudo-automorphisms in dimension $ 3$


Author: Tuyen Trung Truong
Journal: Trans. Amer. Math. Soc. 368 (2016), 727-753
MSC (2010): Primary 37F99, 32H50
DOI: https://doi.org/10.1090/S0002-9947-2014-06340-X
Published electronically: December 16, 2014
MathSciNet review: 3413882
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Abstract: Let $ X$ be a compact Kähler manifold of dimension $ 3$ and let $ f:X\rightarrow X$ be a pseudo-automorphism. Under the mild condition that $ \lambda _1(f)^2>\lambda _2(f)$, we prove the existence of invariant positive closed $ (1,1)$ and $ (2,2)$ currents, and we also discuss the (still open) problem of intersection of such currents. We prove a weak equi-distribution result for Green $ (1,1)$ currents of meromorphic selfmaps, not necessarily algebraic $ 1$-stable, of a compact Kähler manifold of arbitrary dimension and discuss how a stronger equidistribution result may be proved for pseudo-automorphisms in dimension $ 3$. As a byproduct, we show that the intersection of some dynamically related currents is well-defined with respect to our definition here, even though not obviously to be seen so using the usual criteria.


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  • [1] Lucia Alessandrini and Giovanni Bassanelli, Transforms of currents by modifications and 1-convex manifolds, Osaka J. Math. 40 (2003), no. 3, 717-740. MR 2003745 (2004f:32046)
  • [2] Giovanni Bassanelli, A cut-off theorem for plurisubharmonic currents, Forum Math. 6 (1994), no. 5, 567-595. MR 1295153 (95f:32020), https://doi.org/10.1515/form.1994.6.567
  • [3] Turgay Bayraktar, Green currents for meromorphic maps of compact Kähler manifolds, J. Geom. Anal. 23 (2013), no. 2, 970-998. MR 3023864, https://doi.org/10.1007/s12220-012-9315-3
  • [4] Jérémy Blanc, Dynamical degrees of (pseudo)-automorphisms fixing cubic hypersurfaces, Indiana Univ. Math. J. 62 (2013), no. 4, 1143-1164. MR 3179687
  • [5] J.-B. Bost, H. Gillet, and C. Soulé, Heights of projective varieties and positive Green forms, J. Amer. Math. Soc. 7 (1994), no. 4, 903-1027. MR 1260106 (95j:14025), https://doi.org/10.2307/2152736
  • [6] Eric Bedford, S. Cantat, and K.-H. Kim, work in progress, March 2013.
  • [7] Eric Bedford, J. Diller, and K.-H. Kim, work in progress.
  • [8] Eric Bedford and K.-H. Kim, Pseudo-automorphisms without dimension-reducing factors, Manuscript.
  • [9] Eric Bedford and K.-H. Kim, Pseudo-automorphisms of $ 3$-space: periodicities and positive entropy in linear fractional recurrences, arXiv: 1101.1614.
  • [10] Eric Bedford and B. A. Taylor, A new capacity for plurisubharmonic functions, Acta Math. 149 (1982), no. 1-2, 1-40. MR 674165 (84d:32024), https://doi.org/10.1007/BF02392348
  • [11] Serge Cantat, Dynamique des automorphismes des surfaces $ K3$, Acta Math. 187 (2001), no. 1, 1-57 (French). MR 1864630 (2003h:32026), https://doi.org/10.1007/BF02392831
  • [12] Serge Cantat and Igor Dolgachev, Rational surfaces with a large group of automorphisms, J. Amer. Math. Soc. 25 (2012), no. 3, 863-905. MR 2904576, https://doi.org/10.1090/S0894-0347-2012-00732-2
  • [13] Jean-Pierre Demailly, Complex analytic and differential geometry, online book, version of Thursday, 10 September 2009.
  • [14] Jean-Pierre Demailly, Monge-Ampère operators, Lelong numbers and intersection theory, Complex analysis and geometry, Univ. Ser. Math., Plenum, New York, 1993, pp. 115-193. MR 1211880 (94k:32009)
  • [15] Jean-Pierre Demailly, Thomas Peternell, and Michael Schneider, Pseudo-effective line bundles on compact Kähler manifolds, Internat. J. Math. 12 (2001), no. 6, 689-741. MR 1875649 (2003a:32032), https://doi.org/10.1142/S0129167X01000861
  • [16] H. de Thélin and G. de Vigny, Entropy of meromorphic maps and dynamics of birational maps, Memoire de la SMF 122 (2010). MR 2752759 (2011m:37075)
  • [17] Jeffrey Diller and Charles Favre, Dynamics of bimeromorphic maps of surfaces, Amer. J. Math. 123 (2001), no. 6, 1135-1169. MR 1867314 (2002k:32028)
  • [18] Jeffrey Diller, Birational maps, positive currents, and dynamics, Michigan Math. J. 46 (1999), no. 2, 361-375. MR 1704197 (2000e:32024), https://doi.org/10.1307/mmj/1030132416
  • [19] Jeffrey Diller, Romain Dujardin, and Vincent Guedj, Dynamics of meromorphic maps with small topological degree I: from cohomology to currents, Indiana Univ. Math. J. 59 (2010), no. 2, 521-561. MR 2648077 (2012j:32016), https://doi.org/10.1512/iumj.2010.59.4023
  • [20] Jeffrey Diller and Vincent Guedj, Regularity of dynamical Green's functions, Trans. Amer. Math. Soc. 361 (2009), no. 9, 4783-4805. MR 2506427 (2010h:32016), https://doi.org/10.1090/S0002-9947-09-04740-0
  • [21] Tien-Cuong Dinh and Nessim Sibony, Une borne supérieure pour l'entropie topologique d'une application rationnelle, Ann. of Math. (2) 161 (2005), no. 3, 1637-1644 (French, with English summary). MR 2180409 (2006f:32026), https://doi.org/10.4007/annals.2005.161.1637
  • [22] Tien-Cuong Dinh and Nessim Sibony, Regularization of currents and entropy, Ann. Sci. École Norm. Sup. (4) 37 (2004), no. 6, 959-971 (English, with English and French summaries). MR 2119243 (2006c:32045), https://doi.org/10.1016/j.ansens.2004.09.002
  • [23] Tien-Cuong Dinh and Nessim Sibony, Pull-back currents by holomorphic maps, Manuscripta Math. 123 (2007), no. 3, 357-371. MR 2314090 (2008i:32057), https://doi.org/10.1007/s00229-007-0103-5
  • [24] Tien-Cuong Dinh and Nessim Sibony, Green currents for holomorphic automorphisms of compact Kähler manifolds, J. Amer. Math. Soc. 18 (2005), no. 2, 291-312 (electronic). MR 2137979 (2007d:32015), https://doi.org/10.1090/S0894-0347-04-00474-6
  • [25] Tien-Cuong Dinh and Nessim Sibony, Super-potentials of positive closed currents, intersection theory and dynamics, Acta Math. 203 (2009), no. 1, 1-82. MR 2545825 (2011b:32052), https://doi.org/10.1007/s11511-009-0038-7
  • [26] Tien-Cuong Dinh and Nessim Sibony, Super-potentials for currents on compact Kähler manifolds and dynamics of automorphisms, J. Algebraic Geom. 19 (2010), no. 3, 473-529. MR 2629598 (2011f:32072), https://doi.org/10.1090/S1056-3911-10-00549-7
  • [27] Tien-Cuong Dinh and Nessim Sibony, Density of positive closed currents and dynamics of Hénon-type automorphisms of $ \mathbb{C}^k$ (part I), arXiv:1203.5810.
  • [28] Igor Dolgachev and David Ortland, Point sets in projective spaces and theta functions, Astérisque 165 (1988), 210 pp. (1989) (English, with French summary). MR 1007155 (90i:14009)
  • [29] Charles Favre, Note on pull-back and Lelong number of currents, Bull. Soc. Math. France 127 (1999), no. 3, 445-458 (English, with English and French summaries). MR 1724404 (2000i:32057)
  • [30] John Erik Fornæss and Nessim Sibony, Oka's inequality for currents and applications, Math. Ann. 301 (1995), no. 3, 399-419. MR 1324517 (96k:32013), https://doi.org/10.1007/BF01446636
  • [31] John Erik Fornæss and Nessim Sibony, Complex dynamics in higher dimensions, Complex potential theory (Montreal, PQ, 1993) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 439, Kluwer Acad. Publ., Dordrecht, 1994, pp. 131-186. Notes partially written by Estela A. Gavosto. MR 1332961 (96k:32057)
  • [32] Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York, 1978. MR 507725 (80b:14001)
  • [33] Mikhaïl Gromov, On the entropy of holomorphic maps, Enseign. Math. (2) 49 (2003), no. 3-4, 217-235. MR 2026895 (2005h:37097)
  • [34] Vincent Guedj, Decay of volumes under iteration of meromorphic mappings, Ann. Inst. Fourier (Grenoble) 54 (2004), no. 7, 2369-2386 (2005) (English, with English and French summaries). MR 2139697 (2005m:32035)
  • [35] Vincent Guedj, Propriétés ergodiques des applications rationnelles, Quelques aspects des systèmes dynamiques polynomiaux, Panor. Synthèses, vol. 30, Soc. Math. France, Paris, 2010, pp. 97-202 (French, with English and French summaries). MR 2932434
  • [36] P. Lelong, Fonctions plurisousharmoniques et formes différentielles positives, Gordon & Breach, Paris, 1968 (French). MR 0243112 (39 #4436)
  • [37] Michel Meo, Image inverse d'un courant positif fermé par une application analytique surjective, C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), no. 12, 1141-1144 (French, with English and French summaries). MR 1396655 (97d:32013)
  • [38] Keiji Oguiso and Fabio Perroni, Automorphisms of rational manifolds of positive entropy with Siegel disks, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 22 (2011), no. 4, 487-504. MR 2904995
  • [39] Keiji Oguiso, Bimeromorphic automorphism groups of non-projective hyperkähler manifolds--a note inspired by C. T. McMullen, J. Differential Geom. 78 (2008), no. 1, 163-191. MR 2406267 (2009g:32037)
  • [40] Keiji Oguiso, Automorphism groups of Calabi-Yau manifolds of Picard number 2, J. Algebraic Geom. 23 (2014), no. 4, 775-795. MR 3263669
  • [41] Keiji Oguiso, A remark on dynamical degrees of automorphisms of hyperkähler manifolds, Manuscripta Math. 130 (2009), no. 1, 101-111. MR 2533769 (2010m:32021), https://doi.org/10.1007/s00229-009-0271-6
  • [42] Fabio Perroni and De-Qi Zhang, Pseudo-automorphisms of positive entropy on the blowups of products of projective spaces, Math. Ann. 359 (2014), no. 1-2, 189-209. MR 3201898, https://doi.org/10.1007/s00208-013-0992-4
  • [43] Alexander Russakovskii and Bernard Shiffman, Value distribution for sequences of rational mappings and complex dynamics, Indiana Univ. Math. J. 46 (1997), no. 3, 897-932. MR 1488341 (98h:32046), https://doi.org/10.1512/iumj.1997.46.1441
  • [44] Nessim Sibony, Dynamique des applications rationnelles de $ \mathbf {P}^k$, Dynamique et géométrie complexes (Lyon, 1997) Panor. Synthèses, vol. 8, Soc. Math. France, Paris, 1999, pp. ix-x, xi-xii, 97-185 (French, with English and French summaries). MR 1760844 (2001e:32026)
  • [45] Tuyen Trung Truong, The simplicity of the first spectral radius of a meromorphic map, Michigan Math. J. 63 (2014), no. 3, 623-633. MR 3255693, https://doi.org/10.1307/mmj/1409932635
  • [46] Tuyen Trung Truong, Pullback of currents by meromorphic maps, ProQuest LLC, Ann Arbor, MI, Thesis (Ph.D.)-Indiana University, 2012. MR 3054986
  • [47] Y. Yomdin, Volume growth and entropy, Israel J. Math. 57 (1987), no. 3, 285-300. MR 889979 (90g:58008), https://doi.org/10.1007/BF02766215

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Additional Information

Tuyen Trung Truong
Affiliation: Department of Mathematics, Syracuse University, Syracuse, New York 13244
Address at time of publication: School of Mathematics, Korea Institute for Advanced Study, Seoul 130-722, Republic of Korea
Email: tutruong@syr.edu, truong@kias.re.kr

DOI: https://doi.org/10.1090/S0002-9947-2014-06340-X
Keywords: Dynamical degrees, intersection of currents, invariant currents, pseudo-automorphisms, pullback of currents, quasi-potentials.
Received by editor(s): May 8, 2013
Received by editor(s) in revised form: December 3, 2013
Published electronically: December 16, 2014
Article copyright: © Copyright 2014 by the author

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