Real Kodaira surfaces

• Autores: Paola Frediani
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 55, Fasc. 1, 2004, págs. 61-96
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • In this paper we give the topological classification of real primary Kodaira surfaces and we describe in detail the structure of the corresponding moduli space. Moreover, we use the notion of the orbifold fundamental group of a real variety, which was also the main tool in the classification of real hyperelliptic surfaces achieved in [10]. Our first result is that if $(S, \sigma)$ is a real primary Kodaira surface, then the differentiable type of the pair $(S, \sigma)$ is completely determined by the orbifold fundamental group exact sequence. This result allows us to determine all the possible topological types of $(S, \sigma)$. Finally, we show that once we fix the topological type of $(S, \sigma)$ corresponding to a real primary Kodaira surface, the corresponding moduli space is irreducible (and connected).

• Referencias bibliográficas
  • N.L. Alling and N. Greenleaf, Foundations of the Theory of Klein Surfaces, Volume 219 of Lecture Notes in Mathematics, Springer-Verlag, Berlin-Heidelberg-New...
  • W. Barth, C. Peters, and A. Van de Ven, Compact Complex Surfaces, Springer Ergebnisse (3), 4, Springer-Verlag, 1984.
  • A. Beauville, Surfaces Algébriques Complexes, Astérisque 54, Société Mathématique de France, Paris, 1978.
  • C. Borcea, Moduli for Kodaira surfaces, Compositio Math. 52 (1984), 373–380.
  • E. Calabi, Construction and properties of some 6-dimensional almost complex manifolds, Trans. Amer. Math. Soc. 87 (1958), 407–438.
  • F. Catanese, On the moduli spaces of surfaces of general type, J. Differential Geom. 19 (1984), 483–515.
  • F. Catanese, Connected components of moduli spaces, J. Differential Geom. 24 (1986), 395–399.
  • F. Catanese, Moduli spaces of surfaces and real structures, AG/0103071 (2001), to appear in Annals of Math.
  • F. Catanese, Fibred surfaces, varieties isogenous to a product and related moduli spaces, Amer. J. Math. 122 (2000), 1–44.
  • F. Catanese and P. Frediani, Real hyperelliptic surfaces and the orbifold fundamental group, J. Inst. Math. Jussieu 2 (2003), 163–233.
  • C. Ciliberto and C. Pedrini, Real abelian varieties and real algebraic curves, Lectures in real geometry (Madrid, 1994), Gruyter Exp. Math....
  • A. Comessatti, Fondamenti per la geometria sopra le superficie razionali dal punto di vista reale, Math. Annalen 43 (1912), 1–72.
  • A. Comessatti, Sulla connessione delle superficie razionali reali, Annali di Matematica 23 (1914), 215–285.
  • A. Comessatti, Reelle Fragen in der algebraischen Geometrie, Jahresbericht d. Deut. Math. Vereinigung 41 (1932), 107–134.
  • A. Comessatti, Suulle varietà abeliane reali I e II, Ann. Mat. Pura Appl. 2, 4 (1924 and 1926), 67–106 and 27–71.
  • A. Degtyarev, I. Itenberg, and V. Kharlamov, Real Enriques Surfaces, Springer, Lecture Notes in Mathematics, 1746, 2000.
  • A. Degtyarev and V. Kharlamov, Topological classification of real Enriques surfaces, Topology 35 (1996), 711–729.
  • A. Degtyarev and V. Kharlamov, Topological properties of real algebraic varieties: Rokhlin’s way, Russian Math. Surveys 55 (2000), 735–814.
  • A. Degtyarev and V. Kharlamov, On the moduli space of real Enriques surfaces, C. R. Acad. Sci. Paris Sér. I Math. 324 (1997), 317–322.
  • A. Degtyarev and V. Kharlamov, Real rational surfaces are quasi-simple, J. Reine Angew. Math. 551 (2002), 87–99.
  • V. Kharlamov, The topological type of nonsingular surfaces in $\mathbb{RP}^3$ of degree 4, Functional Anal. Appl. 10 (1976), 295–305.
  • V. Kharlamov and Vik. S. Kulikov, On real structures of rigid surfaces, Izv. Ross. Akad. Nauk Ser. Mat. 22 (2002), 133–152; translation in...
  • K. Kodaira and J. Morrow, Complex Manifolds, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1971.
  • K. Kodaira, Complex Manifolds and Deformation of Complex Structures, Springer-Verlag, New York-Berlin, 1986.
  • K. Kodaira, On the structure of compact complex analytic surfaces I, Amer. J. Math. 86 (1964), 751–798.
  • K. Kodaira, On the structure of complex analytic surfaces IV, Amer. J. Math. 90 (1968), 1048–1066.
  • M. Manetti, On the moduli space of diffeomorphic algebraic surfaces, Invent. Math. 143 (2001), 29–76.
  • F. Mangolte, Cycles algébriques sur les surfaces $K3$ réelles, Math. Z. 225 (1997), 559–576.
  • F. Mangolte and J. van Hamel, Algebraic cycles and topology of real Enriques surfaces, Compositio Math. 110 (1998), 215–237.
  • F. Mangolte, Surfaces elliptiques réelles et inégalité de Ragsdale-Viro, Math. Z. 235 (2000), 213–226.
  • V.V. Nikulin, Integer symmetric bilinear forms and some of their geometric applications, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), 111–177.
  • M. Seppälä, Real algebraic curves in the moduli space of complex curves, Compositio Math. 74 (1990), 259–283.
  • M. Seppälä and R. Silhol, Moduli spaces for real algebraic curves and real abelian varieties, Math. Z. 201 (1989), 151–165.
  • R. Silhol, Real Algebraic Surfaces, Lecture Notes in Mathematics 1392, Springer-Verlag, Berlin, 1989.
  • Jean-Yves Welschinger, Real structures on minimal ruled surfaces, Comment. Math. Helv. 78 (2003), 418–446.