Automorphismes loxodromiques de surfaces abéliennes réelles

Zhao, ShengYuan

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 28 (2019) no. 1, pp. 109-127.

On étudie les automorphismes réels de degré dynamique $>1$ des surfaces complexes compactes munies d’une structure réelle. On montre qu’une surface possédant un tel automorphisme est projective. On classifie des surfaces abéliennes réelles en huit types, selon le nombre de composantes connexes de la partie réelle et la simplicité de la surface abélienne complexe sous-jacente. Pour chacun des huit types, on détermine l’ensemble des valeurs de degrés dynamiques qui peuvent être réalisées par des automorphismes réels. On montrera aussi que le degré dynamique minimum sur une surface K3 complexe ne peut pas être réalisée sur une surface K3 réelle.

We study dynamical degree $>1$ real automorphisms of compact complex surfaces with a real structure. We show that a surface with such an automorphism is necessarily projective. We classify real abelian surfaces into eight types, according to the number of connected components of the real part and the simplicity of the underlying complex abelian surface. For each type, we determine the set of values of dynamical degrees which can be realized by real automorphisms. We also prove that the minimum dynamical degree on a complex K3 surface can not be realized on a real K3 surface.

[1] Brandhorst, Simon; González-Alonso, Víctor Automorphisms of minimal entropy on supersingular K3 surfaces, J. Lond. Math. Soc., Volume 97 (2018) no. 2, pp. 282-305 | DOI | MR | Zbl

[2] Cantat, Serge Dynamique des automorphismes des surfaces projectives complexes, C. R. Math. Acad. Sci. Paris, Volume 328 (1999) no. 10, pp. 901-906 | DOI | MR | Zbl

[3] Cantat, Serge Dynamics of automorphisms of compact complex surfaces, Frontiers in complex dynamics (Princeton Mathematics Series), Volume 51, Princeton University Press, 2014, pp. 463-514 | DOI | MR | Zbl

[4] Gromov, Mikhaïl On the entropy of holomorphic maps, Enseign. Math. (2), Volume 49 (2003) no. 3-4, pp. 217-235 | MR | Zbl

[5] McMullen, Curtis T. Dynamics on $K3$ surfaces : Salem numbers and Siegel disks, J. Reine Angew. Math., Volume 545 (2002), pp. 201-233 | DOI | MR | Zbl

[6] McMullen, Curtis T. Dynamics on blowups of the projective plane, Publ. Math., Inst. Hautes Étud. Sci., Volume 105 (2007), pp. 49-89 | DOI | Numdam | MR | Zbl

[7] McMullen, Curtis T. K3 surfaces, entropy and glue, J. Reine Angew. Math., Volume 658 (2011), pp. 1-25 | DOI | MR | Zbl

[8] McMullen, Curtis T. cox.tar, Salem number/Coxeter group/K3 surface package, Harvard Dataverse, 2015 | DOI

[9] McMullen, Curtis T. Automorphisms of projective K3 surfaces with minimum entropy, Invent. Math., Volume 203 (2016) no. 1, pp. 179-215 | DOI | MR | Zbl

[10] Oguiso, Keiji The third smallest Salem number in automorphisms of $K3$ surfaces, Algebraic geometry in East Asia—Seoul 2008 (Advanced Studies in Pure Mathematics), Volume 60, Mathematical Society of Japan, 2010 | DOI | MR | Zbl

[11] Reschke, Paul Salem numbers and automorphisms of complex surfaces, Math. Res. Lett., Volume 19 (2012) no. 2, pp. 475-482 | DOI | MR | Zbl

[12] Reschke, Paul Salem numbers and automorphisms of abelian surfaces, Osaka J. Math., Volume 54 (2017) no. 1, pp. 1-15 | MR | Zbl

[13] Ruppert, Wolfgang M. When is an abelian surface isomorphic or isogeneous to a product of elliptic curves ?, Math. Z., Volume 203 (1990) no. 2, pp. 293-299 | DOI | MR | Zbl

[14] Silhol, Robert Real abelian varieties and the theory of Comessatti, Math. Z., Volume 181 (1982) no. 3, pp. 345-364 | DOI | MR | Zbl

[15] Silhol, Robert Real algebraic surfaces, Lecture Notes in Mathematics, 1392, Springer, 1989 | MR | Zbl

[16] Uehara, Takato Rational surface automorphisms with positive entropy, Ann. Inst. Fourier, Volume 66 (2016) no. 1, pp. 377-432 http://aif.cedram.org/item?id=aif_2016__66_1_377_0 | DOI | Numdam | MR | Zbl

[17] Yomdin, Yosef Volume growth and entropy, Isr. J. Math., Volume 57 (1987) no. 3, pp. 285-300 | DOI | MR | Zbl