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SURFACES OF GENERAL TYPE WITH pg = 1 AND q = 0

  • Park, Heesang (School of Mathematics Korea Institute for Advanced Study) ;
  • Park, Jongil (Department of Mathematical Sciences Seoul National University, Korea Institute for Advanced Study) ;
  • Shin, Dongsoo (Department of Mathematics Chungnam National University, Korea Institute for Advanced Study)
  • Received : 2012.04.04
  • Published : 2013.05.01

Abstract

We construct a new family of simply connected minimal complex surfaces of general type with $p_g$ = 1, $q$ = 0, and $K^2$ = 3, 4, 5, 6, 8 using a $\mathbb{Q}$-Gorenstein smoothing theory.

Keywords

References

  1. W. Barth, K. Hulek, C. Peters, and A. Van de Ven, Compact Complex Surfaces, 2nd ed. Springer-Verlag, Berlin, 2004.
  2. F. Catanese, Surfaces with $K^2$=pg=1 and their period mapping, Algebraic geometry (proc. Summer Meeting, Univ. Copenhagen, Copenhagen, 1978), Lecture Notes in Math. 732 (1979), 1-29.
  3. F. Catanese and O. Debarre, Surfaces with $K^2$=2, pg=1, q=0, J. Reine Angew. Math. 395 (1989), 1-55.
  4. H. Esnault and E. Viehweg, Lectures on vanishing theorems, DMV Seminar 20, Birkhauser Verlag, Basel, 1992.
  5. H. Flenner and M. Zaidenberg, Q-acyclic surfaces and their deformations, Classification of algebraic varieties (L'Aquila, 1992), 143-208, Contemp. Math., 162, Amer. Math. Soc., Providence, RI, 1994.
  6. R. E. Gompf, A new construction of symplectic manifolds, Ann. of Math. 142 (1995), no. 3, 527-595. https://doi.org/10.2307/2118554
  7. J. Keum, Y. Lee, and H. Park, Construction of surfaces of general type from elliptic surfaces via Q-Gorenstein smoothing, Math. Z. 272 (2012), no 3-4, 1243-1257. https://doi.org/10.1007/s00209-012-0985-0
  8. S. Kondo, Enriques surfaces with nite automorphism groups, Japan. J. Math. (N.S.) 12 (1986), no. 2, 191-282. https://doi.org/10.4099/math1924.12.191
  9. V. Kynev, An example of a simply connected surface of general type for which the local Torelli theorem does not hold, C. R. Acad. Bulgare Sci. 30 (1977), no. 3, 323-325.
  10. Y. Lee and J. Park, A simply connected surface of general type with pg = 0 and $K^2$=2, Invent. Math. 170 (2007), no. 3, 483-505. https://doi.org/10.1007/s00222-007-0069-7
  11. Y. Lee, A construction of Horikawa surface via Q-Gorenstein smoothings, Math. Z. 267 (2011), no. 1-2, 15-25. https://doi.org/10.1007/s00209-009-0608-6
  12. B. D. Park, Exotic smooth structures on $3CP^2#n{\overline}{CP^2}$, Part II, Proc. Amer. Math. Soc. 128 (2000), no. 10, 3067-3073.
  13. J. Park, Exotic smooth structures on $3CP^#8\overline{CP^2}$, Bull. London Math. Soc. 39 (2007), no. 1, 95-102. https://doi.org/10.1112/blms/bdl017
  14. H. Park, J. Park, and D. Shin, A simply connected surface of general type with pg = 0 and$K^2$=3, Geom. Topol. 13 (2009), no. 2, 743-767. https://doi.org/10.2140/gt.2009.13.743
  15. H. Park, A simply connected surface of general type with pg = 0 and $K^2=4$, Geom. Topol. 13 (2009), no. 3, 1483-1494. https://doi.org/10.2140/gt.2009.13.1483
  16. A. Stipsicz and Z. Szabo, Small exotic 4-manifolds with $b^+_2$= 3, Bull. London Math. Soc. 38 (2006), no. 3, 501-506. https://doi.org/10.1112/S0024609306018406
  17. A. Todorov, A construction of surfaces with pg = 1, q = 0 and $2{\leq}(K^2){\leq}8$ : Counterexamples of the global Torelli theorem, Invent. Math. 63 (1981), no. 2, 287-304. https://doi.org/10.1007/BF01393879

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