Mathematische Zeitschrift

, Volume 288, Issue 3–4, pp 1195–1253 | Cite as

Infinitesimal deformations of rational surface automorphisms

Article

Abstract

If X is a rational surface without nonzero holomorphic vector field and f is an automorphism of X, we study in several examples the Zariski tangent space of the local deformation space of the pair (Xf).

Mathematics Subject Classification

37F10 14E07 32G05 

Notes

Acknowledgements

The author would like to thank Julie Déserti for many discussions, Igor Dolgachev and Laurent Meerseman for useful comments, Philippe Goutet for the nice pictures and the LaTex editing of the Maple files; and lastly the anonymous referee for his very careful reading and his numerous remarks and comments that led to a considerable improvement of the paper.

References

  1. 1.
    Atiyah, M.F., Bott, R.: A Lefschetz fixed point formula for elliptic complexes. II. Applications. Ann. Math. 2(88), 451–491 (1968)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Barth, W.P., Hulek, K., Peters, C.A.M., Van de Ven, A.: Compact Complex Surfaces, 2nd edn. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vol. 4. Springer, Berlin (2004)Google Scholar
  3. 3.
    Bedford, E., Kim, K.: Periodicities in linear fractional recurrences: degree growth of birational surface maps. Mich. Math. J. 54(3), 647–670 (2006)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bedford, E., Kim, K.: Continuous families of rational surface automorphisms with positive entropy. Math. Ann. 348(3), 667–688 (2010)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Blanc, J.: On the inertia group of elliptic curves in the Cremona group of the plane. Mich. Math. J. 56(2), 315–330 (2008)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Blanc, J.: Dynamical degrees of (pseudo)-automorphisms fixing cubic hypersurfaces. Indiana Univ. Math. J. 62(4), 1143–1164 (2013). doi: 10.1512/iumj.2013.62.5040
  7. 7.
    Blanc, J., Déserti, J.: Embeddings of \({\rm SL}(2,{\mathbb{Z}})\) into the Cremona group. Transform. Groups 17(1), 21–50 (2012)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Brunella, M.: Birational Geometry of Foliations. Publicações Matemáticas do IMPA, Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro (2004)MATHGoogle Scholar
  9. 9.
    Cantat, S.: Dynamics of automorphisms of compact complex surfaces. In: Frontiers in Complex Dynamics, vol. 51, pp. 463–514. Princeton University Press, Princeton (2014)Google Scholar
  10. 10.
    Cantat, S., Dolgachev, I.: Rational surfaces with a large group of automorphisms. J. Am. Math. Soc. 25(3), 863–905 (2012)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Cantat, S., Favre, C.: Symétries birationnelles des surfaces feuilletées. J. Reine Angew. Math. 561, 199–235 (2003)MathSciNetMATHGoogle Scholar
  12. 12.
    Déserti, J., Grivaux, J.: Automorphisms of rational surfaces with positive entropy. Indiana Univ. Math. J. 60(5), 1589–1622 (2011)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Diller, J.: Cremona transformations, surface automorphisms, and plane cubics. Mich. Math. J. 60(2), 409–440 (2011) [With an appendix by I. Dolgachev]Google Scholar
  14. 14.
    Diller, J., Jackson, D., Sommese, A.: Invariant curves for birational surface maps. Trans. Am. Math. Soc. 359(6), 2793–2991 (2007)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Douady, A.: Le problème des modules pour les variétés analytiques complexes (d’après Masatake Kuranishi). In: Séminaire Bourbaki, vol. 9, Exp. No. 277, pp. 7–13. Soc. Math., Paris (1995)Google Scholar
  16. 16.
    Fischer, W., Grauert, H.: Lokal-triviale Familien kompakter komplexer Mannigfaltigkeiten. Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II, 1965, 89–94 (1965)Google Scholar
  17. 17.
    Flenner, H.: Ein Kriterium für die Offenheit der Versalität. Math. Z. 178(4), 449–473 (1981)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Fujiki, A.: Finite automorphism groups of complex tori of dimension two. Publ. Res. Inst. Math. Sci. 24(1), 1–97 (1988)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Fulton, W., MacPherson, R.: A compactification of configuration spaces. Ann. Math. (2) 139(1), 183–225 (1994)Google Scholar
  20. 20.
    Gizatullin, M.H.: Rational \(G\)-surfaces. Izv. Akad. Nauk SSSR Ser. Mat. 44(1), 110–144, 239 (1980)Google Scholar
  21. 21.
    Gizatullin, M.H.: The decomposition, inertia and ramification groups in birational geometry. In: Algebraic Geometry and Its Applications (Yaroslavl’, 1992). Aspects of Mathematics, E25, pp. 39–45. Vieweg, Braunschweig (1994)Google Scholar
  22. 22.
    Griffiths, P., Harris, J.: Principles of Algebraic Geometry. Wiley Classics Library. Wiley, New York (1994) [Reprint of the 1978 original]Google Scholar
  23. 23.
    Grivaux, J.: Parabolic automorphisms of projective surfaces (after M. H. Gizatullin). Mosc. Math. J. 16(2), 275–298 (2016)Google Scholar
  24. 24.
    Gromov, M.: On the entropy of holomorphic maps. Enseign. Math. (2) 49(3–4), 217–235 (2003)Google Scholar
  25. 25.
    Gross, M., Joyce, D., Huybrechts, D.: Calabi–Yau Manifolds and Related Geometries. Universitext. Springer, Berlin (2003). doi: 10.1007/978-3-642-19004-9
  26. 26.
    Harbourne, B.: Rational surfaces with infinite automorphism group and no antipluricanonical curve. Proc. Am. Math. Soc. 99(3), 409–414 (1987)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Harbourne, B.: Anticanonical rational surfaces. Trans. Am. Math. Soc. 349(3), 1191–1208 (1997)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Harris, J., Morrison, I.: Moduli of Curves. Graduate Texts in Mathematics, vol. 187. Springer, New York (1998)Google Scholar
  29. 29.
    Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics, vol. 52. Springer, New York (1977)Google Scholar
  30. 30.
    Huybrechts, D.: Complex Geometry. An Introduction. Universitext. Springer, Berlin (2005)Google Scholar
  31. 31.
    Iskovskikh, V.A., Shafarevich, I.R.: Algebraic surfaces [MR1060325 (91f:14029)]. In: Algebraic Geometry, II. Encyclopaedia of Mathematical Sciences, vol. 35, pp. 127–262. Springer, Berlin (1996)Google Scholar
  32. 32.
    Kodaira, K.: Complex Manifolds and Deformation of Complex Structures. Classics in Mathematics, English edn. Springer, Berlin (2005) [Translated from the 1981 Japanese original by Kazuo Akao]Google Scholar
  33. 33.
    Kodaira, K., Spencer, D.C.: A theorem of completeness for complex analytic fibre spaces. Acta Math. 100, 281–294 (1958)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Kuranishi, M.: On the locally complete families of complex analytic structures. Ann. Math. 75(3), 536 (1962)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Lins Neto, A.: Some examples for the Poincaré and Painlevé problems. Ann. Sci. École Norm. Sup. (4) 35(2), 231–266 (2002)Google Scholar
  36. 36.
    Looijenga, E.: Rational surfaces with an anticanonical cycle. Ann. Math. (2) 114(2), 267–322 (1981)Google Scholar
  37. 37.
    McMullen, C.T.: Dynamics on \(K3\) surfaces: Salem numbers and Siegel disks. J. Reine Angew. Math. 545, 201–233 (2002)MathSciNetMATHGoogle Scholar
  38. 38.
    McMullen, C.T.: Dynamics on blowups of the projective plane. Publ. Math. Inst. Hautes Études Sci. 105, 49–89 (2007)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Meersseman, L.: Foliated structure of the Kuranishi space and isomorphisms of deformation families of compact complex manifolds. Ann. Sci. Éc. Norm. Supér. (4) 44(3), 495–525 (2011)Google Scholar
  40. 40.
    Nagata, M.: On rational surfaces. I. Irreducible curves of arithmetic genus \(0\) or \(1\). Mem. Coll. Sci. Univ. Kyoto Ser. A Math. 32, 351–370 (1960)Google Scholar
  41. 41.
    Puchuri Medina, L.:. Degree of the first integral of a pencil in \({\mathbb{P}}^2\) defined by Lins Neto. Publ. Mat. 57(1), 123–137 (2013)Google Scholar
  42. 42.
    Rim, D.S.: Equivariant \(G\)-structure on versal deformations. Trans. Am. Math. Soc. 257(1), 217–226 (1980)MathSciNetMATHGoogle Scholar
  43. 43.
    Rollenske, S.: The Kuranishi space of complex parallelisable nilmanifolds. J. Eur. Math. Soc. 13(3), 513–531 (2011)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Sernesi, E.: Deformations of Algebraic Schemes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 334. Springer, Berlin (2006)Google Scholar
  45. 45.
    Siebert, B.: Counterexample to the equivariance of versal deformations. http://www.math.uni-hamburg.de/home/siebert/preprints/Gversal.pdf. Preprint
  46. 46.
    Venkata Balaji, T.E.: An Introduction to Families, Deformations and Moduli. Universitätsdrucke Göttingen, Göttingen (2010)CrossRefMATHGoogle Scholar
  47. 47.
    Wavrik, J.J.: Obstructions to the existence of a space of moduli. In: Global Analysis: Papers in Honor of K. Kodaira (PMS-29), pp. 403–414. Princeton University Press, Princeton (1969)Google Scholar
  48. 48.
    Wavrik, J.J.: A theorem of completeness for families of compact analytic spaces. Trans. Am. Math. Soc. 163, 147–155 (1972)MathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    Yomdin, Y.: Volume growth and entropy. Isr. J. Math. 57(3), 285–300 (1987)MathSciNetCrossRefMATHGoogle Scholar
  50. 50.
    Yoshihara, H.: Quotients of abelian surfaces. Publ. Res. Inst. Math. Sci. 31(1), 135–143 (1995)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Deutschland 2017

Authors and Affiliations

  1. 1.CNRS, I2MMarseilleFrance
  2. 2.IHÉSBures-sur-YvetteFrance

Personalised recommendations