Kummer rigidity for hyperkähler automorphisms

Seung uk Jang

doi:10.3934/jmd.2024005

Article Contents

2024, Volume 20: 183-213. Doi: 10.3934/jmd.2024005

This volume Previous Article Periodic points of Prym eigenforms Next Article Invariant measures of the topological flow and measures at infinity on hyperbolic groups

Kummer rigidity for hyperkähler automorphisms

Seung uk Jang

CNRS, IRMAR - UMR 6625, Bâtiment 22/23, Université de Rennes I, 263 avenue du Général Leclerc, 35700 Rennes, France

Received: November 3, 2022

Revised: August 22, 2023

Published online: May 21, 2024

This material is based upon work supported by the National Science Foundation under Grant No. DMS-2005470 and DMS-2305394. The research activities of the author are partially funded by the European Research Council (ERC GOAT 101053021).

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

Abstract

We show that a holomorphic automorphism on a projective hyperkähler manifold that has positive topological entropy and has volume measure as the measure of maximal entropy is necessarily a Kummer example. This partially extends the analogous results in [10,21] for complex surfaces.

A trick with Jensen's inequality is used to show that stable and unstable distributions exhibit uniform rate of contraction and expansion, and with them our hyperkähler manifold is shown to be flat, modulo contracting some loci. A result in [23] then implies that our hyperkähler manifold is birational to a torus quotient, giving the Kummer example structure.

Keywords:

Mathematics Subject Classification: Primary: 14J50, 32H50, 37C40; Secondary: 32U40, 53C26.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	A. Beauville, Variétés Kähleriennes dont la première classe de Chern est nulle, J. Differential Geom., 18 (1983), 755-782. doi: 10.4310/jdg/1214438181.
[2]	A. Beauville, Variétés kählériennes compactes avec $c_1 = 0$, Astérisque, (1985), 181–192.
[3]	E. Bedford and B. A. Taylor, A new capacity for plurisubharmonic functions, Acta Math., 149 (1982), 1-40. doi: 10.1007/BF02392348.
[4]	Y. Benoist, P. Foulon and F. Labourie, Flots d'Anosov à distributions stable et instable différentiables, J. Amer. Math. Soc., 5 (1992), 33-74. doi: 10.2307/2152750.
[5]	L. Bieberbach, Über die Bewegungsgruppen der Euklidischen Räume, Math. Ann., 70 (1911), 297-336. doi: 10.1007/BF01564500.
[6]	L. Bieberbach, Über die Bewegungsgruppen der Euklidischen Räume (Zweite Abhandlung.) Die Gruppen mit einem endlichen Fundamentalbereich, Math. Ann., 72 (1912), 400-412. doi: 10.1007/BF01456724.
[7]	C. Birkar, P. Cascini, C. D. Hacon and J. McKernan, Existence of minimal models for varieties of log general type, J. Amer. Math. Soc., 23 (2010), 405-468. doi: 10.1090/S0894-0347-09-00649-3.
[8]	S. Boucksom, S. Cacciola and A. F. Lopez, Augmented base loci and restricted volumes on normal varieties, Math. Z., 278 (2014), 979-985. doi: 10.1007/s00209-014-1341-3.
[9]	S. Cantat, Dynamics of automorphisms of compact complex surfaces, in Frontiers in Complex Dynamics, 463–514, Princeton Math. Ser., 51, Princeton Univ. Press, Princeton, NJ, 2014.
[10]	S. Cantat and C. Dupont, Automorphisms of surfaces: Kummer rigidity and measure of maximal entropy, J. Eur. Math. Soc. (JEMS), 22 (2020), 1289-1351. doi: 10.4171/JEMS/946.
[11]	B. Claudon, P. Graf, H. Guenancia and P. Naumann, Kähler spaces with zero first Chern class: Bochner principle, Albanese map and fundamental groups, J. Reine Angew. Math., 786 (2022), 245-275. doi: 10.1515/crelle-2022-0001.
[12]	T. C. Collins and V. Tosatti, Kähler currents and null loci, Invent. Math., 202 (2015), 1167-1198. doi: 10.1007/s00222-015-0585-9.
[13]	M. A. de Gosson, Symplectic Methods in Harmonic Analysis and in Mathematical Physics, Pseudo-Differential Operators. Theory and Applications, vol. 7, Birkhäuser/Springer Basel AG, Basel, 2011. doi: 10.1007/978-3-7643-9992-4.
[14]	H. de Thélin, Sur les exposants de Lyapounov des applications méromorphes, Invent. Math., 172 (2008), 89-116. doi: 10.1007/s00222-007-0095-5.
[15]	H. De Thélin and T.-C. Dinh, Dynamics of automorphisms on compact Kähler manifolds, Adv. Math., 229 (2012), 2640-2655. doi: 10.1016/j.aim.2012.01.014.
[16]	J.-P. Demailly and M. Paun, Numerical characterization of the Kähler cone of a compact Kähler manifold, Ann. of Math. (2), 159 (2004), 1247-1274. doi: 10.4007/annals.2004.159.1247.
[17]	T.-C. Dinh and N. Sibony, Green currents for holomorphic automorphisms of compact Kähler manifolds, J. Amer. Math. Soc., 18 (2005), 291-312. doi: 10.1090/S0894-0347-04-00474-6.
[18]	T.-C. Dinh and N. Sibony, Super-potentials for currents on compact Kähler manifolds and dynamics of automorphisms, J. Algebraic Geom., 19 (2010), 473-529. doi: 10.1090/S1056-3911-10-00549-7.
[19]	M. Fekete, Über die verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten, Math. Z., 17 (1923), 228-249. doi: 10.1007/BF01504345.
[20]	S. Filip, Notes on the multiplicative ergodic theorem, Ergodic Theory Dynam. Systems, 39 (2019), 1153-1189. doi: 10.1017/etds.2017.68.
[21]	S. Filip and V. Tosatti, Kummer rigidity for K3 surface automorphisms via Ricci-flat metrics, Amer. J. Math., 143 (2021), 1431-1462. doi: 10.1353/ajm.2021.0036.
[22]	É. Ghys, Holomorphic Anosov systems, Invent. Math., 119 (1995), 585-614. doi: 10.1007/BF01245193.
[23]	D. Greb, S. Kebekus and T. Peternell, Étale fundamental groups of Kawamata log terminal spaces, flat sheaves, and quotients of abelian varieties, Duke Math. J., 165 (2016), 1965-2004. doi: 10.1215/00127094-3450859.
[24]	M. Gromov, On the entropy of holomorphic maps, Enseign. Math. (2), 49 (2003), 217-235.
[25]	M. Gross, D. Huybrechts and D. Joyce, Calabi-Yau Manifolds and Related Geometries, Lectures from the Summer School held in Nordfjordeid, June 2001, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-642-19004-9.
[26]	R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977.
[27]	T. Kato, Perturbation Theory for Linear Operators, reprint of the 1980 edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995.
[28]	A. Katok and B. Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187.
[29]	Y. Kawamata, Crepant blowing-up of 3-dimensional canonical singularities and its application to degenerations of surfaces, Ann. of Math. (2), 127 (1988), 93-163. doi: 10.2307/1971417.
[30]	Y. Kawamata, K. Matsuda and K. Matsuki, Introduction to the minimal model problem, in Algebraic Geometry, Sendai, 1985, 283–360, Adv. Stud. Pure Math., 10, North-Holland, Amsterdam, 1987. doi: 10.2969/aspm/01010283.
[31]	J. Kollár and S. Mori, Birational Geometry of Algebraic Varieties, with the collaboration of C. H. Clemens and A. Corti, translated from the 1998 Japanese original, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998. doi: 10.1017/CBO9780511662560.
[32]	F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. Ⅱ. Relations between entropy, exponents and dimension, Ann. of Math. (2), 122 (1985), 540-574. doi: 10.2307/1971329.
[33]	J. M. Lee, Introduction to Smooth Manifolds, second edition, Graduate Texts in Mathematics, vol. 218, Springer, New York, 2013.
[34]	F. Lo Bianco, Dynamique des Transformations Birationnelles des Vari'et'es Hyperkähleriennes: Feuilletages et Fibrations Invariantes, Ph.D. thesis, Université de Rennes 1, Thése de doctorat dirigée par Cantat, Serge, Mathématiques et applications Rennes 1, 2017.
[35]	R. Mañé, Ergodic Theory and Differentiable Dynamics, translated from the Portuguese by Silvio Levy, Ergeb. Math. Grenzgeb. (3), vol. 8, Springer-Verlag, Berlin, 1987. doi: 10.1007/978-3-642-70335-5.
[36]	K. Oguiso, Salem polynomials and birational transformation groups for hyper-Kähler manifolds, Sugaku, 59 (2007), 1-23.
[37]	K. Oguiso, A remark on dynamical degrees of automorphisms of hyperkähler manifolds, Manuscripta Mathematica, 130 (2009), 101-111. doi: 10.1007/s00229-009-0271-6.
[38]	D. Ruelle, Ergodic theory of differentiable dynamical systems, Inst. Hautes Études Sci. Publ. Math., no. 50 (1979), 27–58.
[39]	W. P. Thurston, The Geometry and Topology of Three-Manifolds. Vol. IV, edited and with a preface by Steven P. Kerckhoff and a chapter by J. W. Milnor, American Mathematical Society, Providence, RI, 2022.
[40]	M. S. Verbitsky, Cohomology of Compact Hyperkaehler Manifolds, Ph.D. Thesis, Harvard University, ProQuest LLC, Ann Arbor, MI, 1995.
[41]	J. Wierzba, Contractions of symplectic varieties, J. Algebraic Geom., 12 (2003), 507-534. doi: 10.1090/S1056-3911-03-00318-7.
[42]	Y. Yomdin, Volume growth and entropy, Israel J. Math., 57 (1987), 285-300. doi: 10.1007/BF02766215.