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.
On étudie les singularités obtenues en contractant le diviseur maximal des surfaces (non kählerienne) qui contiennent des coquilles sphériques globales. Ces singularités sont de genre 1 ou 2, peuvent être -Gorenstein, numériquement Gorenstein ou de Gorenstein. On définit une famille de polynômes qui dépendent de la configuration des courbes rationnelles pour calculer les discriminants des formes quadratiques associées à ces singularités. Un invariant topologique multiplicatif, défini à partir des arbres du graphe détermine le coefficient de torsion des 1-formes holomorphes tordues qui ne s’annulent pas sur le complémentaire du point singulier.
@article{AFST_2011_6_20_1_15_0, author = {Dloussky, Georges}, title = {Quadratic forms and singularities of genus one or two}, journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques}, pages = {15--69}, publisher = {Universit\'e Paul Sabatier, Institut de math\'ematiques}, address = {Toulouse}, volume = {Ser. 6, 20}, number = {1}, year = {2011}, doi = {10.5802/afst.1285}, mrnumber = {2829832}, language = {en}, url = {http://www.numdam.org/articles/10.5802/afst.1285/} }
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 PY - 2011 SP - 15 EP - 69 VL - 20 IS - 1 PB - Université Paul Sabatier, Institut de mathématiques PP - Toulouse UR - http://www.numdam.org/articles/10.5802/afst.1285/ DO - 10.5802/afst.1285 LA - en ID - AFST_2011_6_20_1_15_0 ER -
%0 Journal Article %A Dloussky, Georges %T Quadratic forms and singularities of genus one or two %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2011 %P 15-69 %V 20 %N 1 %I Université Paul Sabatier, Institut de mathématiques %C Toulouse %U http://www.numdam.org/articles/10.5802/afst.1285/ %R 10.5802/afst.1285 %G en %F AFST_2011_6_20_1_15_0
Dloussky, Georges. Quadratic forms and singularities of genus one or two. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 20 (2011) no. 1, pp. 15-69. doi : 10.5802/afst.1285. http://www.numdam.org/articles/10.5802/afst.1285/
[1] Barth (W.), Hulek (K.), Peters (C.), and Van de Ven (A.).— Compact Complex Surfaces, Springer, Heidelberg, Second Edition (2004). | MR | Zbl
[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). | Numdam | MR | Zbl
[4] Dloussky (G.).— Sur la classification des germes d’applications holomorphes contractantes. Math. Ann. 280, p. 649-661 (1988). | MR | Zbl
[5] Dloussky (G.).— Une construction élémentaire des surfaces d’Inoue-Hirzebruch. Math. Ann. 280, p. 663-682 (1988). | MR | Zbl
[6] Dloussky (G.).— On surfaces of class VII with numerically anticanonical divisor, Am. J. Math. 128(3), p. 639-670 (2006). | MR | Zbl
[7] Dloussky (G.), Oeljeklaus (K.).— Vector fields and foliations on compact surfaces of class VII. Ann. Inst. Fourier 49, p. 1503-1545 (1999). | Numdam | MR | Zbl
[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). | MR | Zbl
[9] Dloussky (G.), Oeljeklaus (K.), Toma (M.).— Class VII surfaces with curves.Tohoku Math. J. 55, p. 283-309 (2003). | MR | Zbl
[10] Enoki (I.).— Surfaces of class VII with curves. Tôhoku Math. J. 33, p. 453-492 (1981). | MR | Zbl
[11] Favre (Ch.).— Classification of -dimensional contracting rigid germs, Jour. Math. Pures Appl. 79, p. 475-514 (2000). | MR | Zbl
[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). | MR | Zbl
[14] Inoue (M.).— New surfaces with no meromorphic functions II. Complex Analysis and Alg. Geom. p. 91-106. Iwanami Shoten Pb. (1977). | MR | Zbl
[15] Karras (U.).— Deformations of cusps singularities. Proc. of Symp. in pure Math. 30, p. 37-44, AMS, Providence (1977). | MR | Zbl
[16] Kato (Ma.).— Compact complex manifolds containing “global spherical shells” I Proc. of the Int. Symp. Alg. Geometry, Kyoto (1977) Iwanami Shoten Publ. | MR | Zbl
[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). | MR | Zbl
[18] Laufer (H.).— On minimally elliptic singularities. Amer. J. of Math. 99, p. 1257-1295, (1977). | MR | Zbl
[19] Looijenga (E.) & Wahl (J.).— Quadratic functions and smoothing surface singularities. Topology 25, p. 261-291 (1986). | MR | Zbl
[20] Mérindol (J.Y.).— Surfaces normales dont le faisceau dualisant est trivial. C.R.A.S. 293, p. 417-420 (1981). | MR | Zbl
[21] Nakamura (I.).— On surfaces of class with curves. Proc. Japan Academy 58A, p. 380-383 (1982) | MR | Zbl
[22] Nakamura (I.).— On surfaces of class with curves. Invent. Math. 78, p. 393-443 (1984). | EuDML | MR | Zbl
[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). | MR | Zbl
[24] Nakamura (I.).— On the equations . Advanced Studies in pure Math. 8, Complex An. Singularities, p. 281-313 (1986). | MR | Zbl
[25] Nakamura (I.).— Inoue-Hirzebruch surfaces and a duality of hyperbolic unimodular singularities I. Math. Ann. 252, p. 221-235 (1980). | EuDML | MR | Zbl
[26] Oeljeklaus (K.), Toma (M.).— Logarithmic moduli spaces for surfaces of class VII, Math. Ann. 341, p. 323-345 (2008). | MR | Zbl
[27] Pinkham (H.).— Singularités rationnelles de surfaces. Appendice. Séminaire sur les singularités des surfaces. Lecture Notes 777. Springer-Verlag (1980). | MR | Zbl
[28] Ribenboim (R.).— Polynomials whose values are powers. J. für die reine und ang. Math. 268/269, p. 34-40 (1974). | EuDML | MR | Zbl
[29] Riemenschneider (O.).— Familien komplexer Räume mit streng pseudokonvexer spezieller Faser. Comment. Math. Helvetici 39(51) p. 547-565 (1976). | EuDML | MR | Zbl
[30] Sakai (F.).— Enriques classification of normal Gorenstein surfaces. Am. J. of Math. 104, p. 1233-1241 (1981). | MR | Zbl
[31] Serre (J.P.).— Cours d’arithmétique Presses Universitaires de France (1970). | MR | Zbl
[32] Teleman (A.).— Projectively flat surfaces and Bogomolov’s theorem on class – surfaces, Int. J. Math., Vol.5, No 2, p. 253-264, (1994). | MR | Zbl
Cited by Sources: