Quadratic forms and singularities of genus one or two

Georges Dloussky[1]

  • [1] Centre de Mathématiques et d’Informatique, Université de Provence, 39 rue F. Joliot-Curie 13453 Marseille Cedex 13, France.

Annales de la faculté des sciences de Toulouse Mathématiques (2011)

  • Volume: 20, Issue: 1, page 15-69
  • ISSN: 0240-2963

Abstract

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We study singularities obtained by the contraction of the maximal divisor in compact (non-kählerian) surfaces which contain global spherical shells. These singularities are of genus 1 or 2, may be -Gorenstein, numerically Gorenstein or Gorenstein. A family of polynomials depending on the configuration of the curves computes the discriminants of the quadratic forms of these singularities. We introduce a multiplicative branch topological invariant which determines the twisting coefficient of a non-vanishing holomorphic 1-form on the complement of the singular point.

How to cite

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Dloussky, Georges. "Quadratic forms and singularities of genus one or two." Annales de la faculté des sciences de Toulouse Mathématiques 20.1 (2011): 15-69. <http://eudml.org/doc/219761>.

@article{Dloussky2011,
abstract = {We study singularities obtained by the contraction of the maximal divisor in compact (non-kählerian) surfaces which contain global spherical shells. These singularities are of genus 1 or 2, may be $\mathbb\{Q\}$-Gorenstein, numerically Gorenstein or Gorenstein. A family of polynomials depending on the configuration of the curves computes the discriminants of the quadratic forms of these singularities. We introduce a multiplicative branch topological invariant which determines the twisting coefficient of a non-vanishing holomorphic 1-form on the complement of the singular point.},
affiliation = {Centre de Mathématiques et d’Informatique, Université de Provence, 39 rue F. Joliot-Curie 13453 Marseille Cedex 13, France.},
author = {Dloussky, Georges},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {minimal compact complex surfaces in class ; global spherical shells; singularities; -Gorenstein; numerically Gorenstein; twisting coefficient},
language = {eng},
month = {1},
number = {1},
pages = {15-69},
publisher = {Université Paul Sabatier, Toulouse},
title = {Quadratic forms and singularities of genus one or two},
url = {http://eudml.org/doc/219761},
volume = {20},
year = {2011},
}

TY - JOUR
AU - Dloussky, Georges
TI - Quadratic forms and singularities of genus one or two
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2011/1//
PB - Université Paul Sabatier, Toulouse
VL - 20
IS - 1
SP - 15
EP - 69
AB - We study singularities obtained by the contraction of the maximal divisor in compact (non-kählerian) surfaces which contain global spherical shells. These singularities are of genus 1 or 2, may be $\mathbb{Q}$-Gorenstein, numerically Gorenstein or Gorenstein. A family of polynomials depending on the configuration of the curves computes the discriminants of the quadratic forms of these singularities. We introduce a multiplicative branch topological invariant which determines the twisting coefficient of a non-vanishing holomorphic 1-form on the complement of the singular point.
LA - eng
KW - minimal compact complex surfaces in class ; global spherical shells; singularities; -Gorenstein; numerically Gorenstein; twisting coefficient
UR - http://eudml.org/doc/219761
ER -

References

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  1. Barth (W.), Hulek (K.), Peters (C.), and Van de Ven (A.).— Compact Complex Surfaces, Springer, Heidelberg, Second Edition (2004). Zbl1036.14016MR2030225
  2. Demailly (J.P.).— Complex Analytic and Differential Geometry (1997). http://www-fourier.ujf-grenoble.fr/~demailly/books.html. 
  3. Dloussky (G.).— Structure des surfaces de Kato. Mémoire de la S.M.F 112. (1984). Zbl0543.32012MR763959
  4. Dloussky (G.).— Sur la classification des germes d’applications holomorphes contractantes. Math. Ann. 280, p. 649-661 (1988). Zbl0677.32004MR939924
  5. Dloussky (G.).— Une construction élémentaire des surfaces d’Inoue-Hirzebruch. Math. Ann. 280, p. 663-682 (1988). Zbl0617.14025MR939925
  6. Dloussky (G.).— On surfaces of class VII with numerically anticanonical divisor, Am. J. Math. 128(3), p. 639-670 (2006). Zbl1102.32007MR2230920
  7. Dloussky (G.), Oeljeklaus (K.).— Vector fields and foliations on compact surfaces of class VII. Ann. Inst. Fourier 49, p. 1503-1545 (1999). Zbl0978.32021MR1723825
  8. Dloussky (G.), Oeljeklaus (K.).— Surfaces de la classe VII et automorphismes de Hénon. C.R.A.S. 328, série I, p. 609-612 (1999). Zbl0945.32005MR1679974
  9. Dloussky (G.), Oeljeklaus (K.), Toma (M.).— Class VII surfaces with curves.Tohoku Math. J. 55, p. 283-309 (2003). Zbl1034.32012MR1979500
  10. Enoki (I.).— Surfaces of class VII with curves. Tôhoku Math. J. 33, p. 453-492 (1981). Zbl0476.14013MR643229
  11. Favre (Ch.).— Classification of -dimensional contracting rigid germs, Jour. Math. Pures Appl. 79, p. 475-514 (2000). Zbl0983.32023MR1759437
  12. Favre (Ch.).— Dynamique des applications rationnelles. Thèse pour le grade de Docteur en Sciences. Université de Paris XI Orsay (2000). http://tel.archives-ouvertes.fr/tel-00003577/fr/ 
  13. Hirzebruch (F.).— Hilbert modular surfaces. L’enseignement Math. 19, p. 183-281 (1973). Zbl0285.14007MR393045
  14. Inoue (M.).— New surfaces with no meromorphic functions II. Complex Analysis and Alg. Geom. p. 91-106. Iwanami Shoten Pb. (1977). Zbl0365.14011MR442297
  15. Karras (U.).— Deformations of cusps singularities. Proc. of Symp. in pure Math. 30, p. 37-44, AMS, Providence (1977). Zbl0352.14007MR472811
  16. Kato (Ma.).— Compact complex manifolds containing “global spherical shells” I Proc. of the Int. Symp. Alg. Geometry, Kyoto (1977) Iwanami Shoten Publ. Zbl0421.32010MR440076
  17. Kodaira (K.).— On the structure of compact complex analytic surfaces I, II. Am. J. of Math. vol.86, p. 751-798 (1964); vol.88, p. 682-721 (1966). Zbl0137.17501MR187255
  18. Laufer (H.).— On minimally elliptic singularities. Amer. J. of Math. 99, p. 1257-1295, (1977). Zbl0384.32003MR568898
  19. Looijenga (E.) & Wahl (J.).— Quadratic functions and smoothing surface singularities. Topology 25, p. 261-291 (1986). Zbl0615.32014MR842425
  20. Mérindol (J.Y.).— Surfaces normales dont le faisceau dualisant est trivial. C.R.A.S. 293, p. 417-420 (1981). Zbl0482.14012MR641100
  21. Nakamura (I.).— On surfaces of class with curves. Proc. Japan Academy 58A, p. 380-383 (1982) Zbl0519.32017MR694939
  22. Nakamura (I.).— On surfaces of class with curves. Invent. Math. 78, p. 393-443 (1984). Zbl0575.14033MR768987
  23. Nakamura (I.).— On surfaces of class with Global Spherical Shells. Proc. of the Japan Acad. 59, Ser. A, No 2, p. 29-32 (1983). Zbl0536.14022MR696743
  24. Nakamura (I.).— On the equations . Advanced Studies in pure Math. 8, Complex An. Singularities, p. 281-313 (1986). Zbl0643.14003MR894299
  25. Nakamura (I.).— Inoue-Hirzebruch surfaces and a duality of hyperbolic unimodular singularities I. Math. Ann. 252, p. 221-235 (1980). Zbl0425.14010MR593635
  26. Oeljeklaus (K.), Toma (M.).— Logarithmic moduli spaces for surfaces of class VII, Math. Ann. 341, p. 323-345 (2008). Zbl1144.32004MR2385660
  27. Pinkham (H.).— Singularités rationnelles de surfaces. Appendice. Séminaire sur les singularités des surfaces. Lecture Notes 777. Springer-Verlag (1980). Zbl0459.14009MR579026
  28. Ribenboim (R.).— Polynomials whose values are powers. J. für die reine und ang. Math. 268/269, p. 34-40 (1974). Zbl0299.12103MR364202
  29. Riemenschneider (O.).— Familien komplexer Räume mit streng pseudokonvexer spezieller Faser. Comment. Math. Helvetici 39(51) p. 547-565 (1976). Zbl0338.32013MR437808
  30. Sakai (F.).— Enriques classification of normal Gorenstein surfaces. Am. J. of Math. 104, p. 1233-1241 (1981). Zbl0512.14022MR681736
  31. Serre (J.P.).— Cours d’arithmétique Presses Universitaires de France (1970). Zbl0376.12001MR255476
  32. Teleman (A.).— Projectively flat surfaces and Bogomolov’s theorem on class – surfaces, Int. J. Math., Vol.5, No 2, p. 253-264, (1994). Zbl0803.53038MR1266285

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