Normal surface singularities admitting contracting automorphisms
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 23 (2014) no. 4, pp. 797-828.

Nous montrons qu’une singularité normale de surface complexe admettant un automorphisme contractant est quasi-homogène. Nous décrivons aussi la géométrie de la surface complexe compacte obtenue comme espace des orbites d’un tel automorphisme contractant.

We show that a complex normal surface singularity admitting a contracting automorphism is necessarily quasihomogeneous. We also describe the geometry of a compact complex surface arising as the orbit space of such a contracting automorphism.

@article{AFST_2014_6_23_4_797_0,
     author = {Favre, Charles and Ruggiero, Matteo},
     title = {Normal surface singularities admitting contracting automorphisms},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {797--828},
     publisher = {Universit\'e Paul Sabatier, Institut de math\'ematiques},
     address = {Toulouse},
     volume = {Ser. 6, 23},
     number = {4},
     year = {2014},
     doi = {10.5802/afst.1425},
     zbl = {1305.14019},
     mrnumber = {3270424},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/afst.1425/}
}
TY  - JOUR
AU  - Favre, Charles
AU  - Ruggiero, Matteo
TI  - Normal surface singularities admitting contracting automorphisms
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2014
DA  - 2014///
SP  - 797
EP  - 828
VL  - Ser. 6, 23
IS  - 4
PB  - Université Paul Sabatier, Institut de mathématiques
PP  - Toulouse
UR  - http://archive.numdam.org/articles/10.5802/afst.1425/
UR  - https://zbmath.org/?q=an%3A1305.14019
UR  - https://www.ams.org/mathscinet-getitem?mr=3270424
UR  - https://doi.org/10.5802/afst.1425
DO  - 10.5802/afst.1425
LA  - en
ID  - AFST_2014_6_23_4_797_0
ER  - 
Favre, Charles; Ruggiero, Matteo. Normal surface singularities admitting contracting automorphisms. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 23 (2014) no. 4, pp. 797-828. doi : 10.5802/afst.1425. http://archive.numdam.org/articles/10.5802/afst.1425/

[1] Boucksom (S.), de Fernex (T.), Favre (C.).— The volume of an isolated singularity, Duke Math. J., 161(8), p. 1455-1520 (2012). | MR 2931273 | Zbl 1251.14026

[2] Barth (W. P.), Hulek (K.), Peters (C. A. M.), Antonius Van de Ven.— Compact complex surfaces, volume 4 of Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], Springer-Verlag, Berlin, second edition (2004). | MR 2030225 | Zbl 1036.14016

[3] Bundgaard (S.), Nielsen (J.).— On normal subgroups with finite index in F-groups, Mat. Tidsskr. B., 1951, p. 56-58 (1951). | MR 48447 | Zbl 0044.25403

[4] Cantat (S.).— Dynamique des automorphismes des surfaces projectives complexes, C. R. Acad. Sci. Paris Sér. I Math., 328(10), p. 901-906 (1999). | MR 1689873 | Zbl 0943.37021

[5] Camacho (C.), Movasati (H.), Bruno (B.).— The moduli of quasi-homogeneous Stein surface singularities. J. Geom. Anal., 19(2), p. 244-260 (2009). | MR 2481961 | Zbl 1186.32007

[6] Demailly (J.-P.).— Monge-Ampère operators, Lelong numbers and intersection theory, In Complex analysis and geometry, Univ. Ser. Math., p. 115-193. Plenum, New York (1993). | MR 1211880 | Zbl 0792.32006

[7] Fox (R. H.).— On Fenchel’s conjecture about F-groups. Mat. Tidsskr. B., 1952, p. 61-65 (1952). | MR 53937 | Zbl 0049.15404

[8] Furuta (M.), Steer (B.).— Seifert fibred homology 3-spheres and the Yang-Mills equations on Riemann surfaces with marked points, Adv. Math., 96(1), p. 38-102 (1992). | MR 1185787 | Zbl 0769.58009

[9] Gizatullin (M. H.).— Rational G-surfaces, Izv. Akad. Nauk SSSR Ser. Mat., 44(1), p. 110-144, 239 (1980). | MR 563788 | Zbl 0428.14022

[10] Grauert (H.).— Über Modifikationen und exzeptionelle analytische Mengen, Math. Ann., 146, p. 331-368 (1962). | EuDML 160940 | MR 137127 | Zbl 0173.33004

[11] Holmann (H.).— Quotientenräume komplexer Mannigfaltigkeiten nach komplexen Lieschen Automorphismengruppen, Math. Ann., 139, p. 383-402 (1960). | EuDML 160755 | MR 115187 | Zbl 0142.05001

[12] Höring (A.), Peternell (T.).— Non-algebraic compact Kähler threefolds admitting endomorphisms, Sci. China Math., 54(8), p. 1635-1664 (2011). | MR 2824964 | Zbl 1241.32014

[13] Kato (M.).— Compact complex surfaces containing global strongly pseudoconvex hypersurfaces, Tôhoku Math. J. (2), 31(4), p. 537-547 (1979). | MR 558683 | Zbl 0428.32012

[14] Laufer (H. B.).— Normal two-dimensional singularities, Princeton University Press, Princeton, N.J. Annals of Mathematics Studies, No. 71 (1971). | MR 320365 | Zbl 0245.32005

[15] Milnor (J.).— On the concept of attractor, Comm. Math. Phys., 99(2), p. 177-195 (1985). | MR 790735 | Zbl 0595.58028

[16] Milnor (J.).— Dynamics in one complex variable, volume 160 of Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, third edition (2006). | MR 2193309 | Zbl 1085.30002

[17] Müller (G.).— Actions of complex Lie groups on analytic C-algebras, Monatsh. Math., 103(3), p. 221-231 (1987). | EuDML 178327 | MR 894172 | Zbl 0624.32007

[18] Müller (G.).— Symmetries of surface singularities, J. London Math. Soc. (2), 59(2), p. 491-506 (1999). | MR 1709181 | Zbl 0923.14029

[19] Müller (G.).— Resolution of weighted homogeneous surface singularities, In Resolution of singularities (Obergurgl, 1997), volume 181 of Progr. Math., p. 507-517. Birkhäuser, Basel (2000). | MR 1748632 | Zbl 1024.14006

[20] Nakayama (N.).— On complex normal projective surfaces admitting non-isomorphic surjective endomorphisms, Preprint.

[21] Nakayama (N.), Zhang (D-Q).— Building blocks of étale endomorphisms of complex projective manifolds, Proc. Lond. Math. Soc. (3), 99(3), p. 725-756 (2009). | MR 2551469 | Zbl 1185.14012

[22] Nakayama (N.), Zhang (D-Q).— Polarized endomorphisms of complex normal varieties, Math. Ann., 346(4), p. 991-1018 (2010). | MR 2587100 | Zbl 1189.14043

[23] Oda (T.).— Convex bodies and algebraic geometry, volume 15 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Springer-Verlag, Berlin. An introduction to the theory of toric varieties, Translated from the Japanese (1988). | EuDML 203658 | MR 922894 | Zbl 0628.52002

[24] Orlik (P.), Wagreich (P.).— Isolated singularities of algebraic surfaces with C * action, Ann. of Math. (2), 93, p. 205-228 (1971). | MR 284435 | Zbl 0212.53702

[25] Pinkham (H. C.).— Automorphisms of cusps and Inoue-Hirzebruch surfaces, Compositio Math., 52(3), p. 299-313 (1984). | EuDML 89666 | Numdam | MR 756724 | Zbl 0573.14015

[26] Rosay (J.-P.), Rudin (W.).— Holomorphic maps from C n to C n , Trans. Amer. Math. Soc., 310(1), p. 47-86 (1988). | MR 929658 | Zbl 0708.58003

[27] Rashkovskii (A.), Sigurdsson (R.).— Green functions with singularities along complex spaces, Internat. J. Math., 16(4), p. 333-355 (2005). | MR 2133260 | Zbl 1085.32018

[28] Ross (J.), Thomas (R.).— Weighted projective embeddings, stability of orbifolds, and constant scalar curvature Käher metrics, J. Differential Geom., 88(1), p. 109-159 (2011). | MR 2819757 | Zbl 1244.32013

[29] Sankaran (G. K.).— Higher-dimensional analogues of Inoue-Hirzebruch surfaces, Math. Ann., 276(3), p. 515-528 (1987). | EuDML 164216 | MR 875345 | Zbl 0595.14031

[30] Satake (I.).— The Gauss-Bonnet theorem for V-manifolds, J. Math. Soc. Japan, 9, p. 464-492 (1957). | MR 95520 | Zbl 0080.37403

[31] Sánchez-Bringas (F.).— Normal forms of invariant vector fields under a finite group action, Publ. Mat., 37(1), p. 75-82 (1993). | EuDML 41521 | MR 1240923 | Zbl 0872.58057

[32] Scott (P.).— The geometries of 3-manifolds. Bull, London Math. Soc., 15(5), p. 401-487 (1983). | MR 705527 | Zbl 0561.57001

[33] Selberg (A.).— On discontinuous groups in higher-dimensional symmetric spaces, In Contributions to function theory (internat. Colloq. Function Theory, Bombay, 1960), p. 147-164. Tata Institute of Fundamental Research, Bombay (1960). | MR 130324 | Zbl 0201.36603

[34] Scheja (G.), Wiebe (H.).— Zur Chevalley-Zerlegung von Derivationen, Manuscripta Math., 33(2), p. 159-176 (1980/81). | EuDML 154746 | MR 597817 | Zbl 0511.13015

[35] Tsuchihashi (H.).— Higher-dimensional analogues of periodic continued fractions and cusp singularities, Tohoku Math. J. (2), 35(4), p. 607-639 (1983). | MR 721966 | Zbl 0585.14004

[36] Wagreich (P.).— The structure of quasihomogeneous singularities, In Singularities, Part 2 (Arcata, Calif., 1981), volume 40 of Proc. Sympos. Pure Math., p. 593-611. Amer. Math. Soc., Providence, RI (1983). | MR 713284 | Zbl 0545.14028

[37] Wahl (J.).— A characteristic number for links of surface singularities, J. Amer. Math. Soc., 3(3), p. 625-637 (1990). | MR 1044058 | Zbl 0743.14026

[38] Zhang (D-Q).— Algebraic varieties with automorphism groups of maximal rank, Math. Ann., 355(1), p. 131-146 (2013). | MR 3004578 | Zbl 1262.32019

Cité par Sources :