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Quadratic forms and singularities of genus one or two. (English) Zbl 1219.14022
The article under review contributes to the structure theory of minimal compact complex surfaces $$S$$ in class VII$$_0^+$$ ($$b_1(S)=1, \kappa(S)=-\infty, b_2(S)=:n>0$$) with global spherical shells. The latter means that there exists a biholomorphic embedding $$\varphi:U\rightarrow S$$ of an open neighborhood $$U\subset\mathbb C^2\backslash\{0\}$$ of the 3-sphere $$S^3$$ such that $$S\backslash\varphi(S^3)$$ is connected. The author studies the (normal) singularities obtained by blowing down the maximal divisor $$D=\sum_{i=0}^{n-1}D_i$$, the sum of the $$n$$ irreducible rational curves $$D_i$$ on $$S$$. These singularities are of genus $$1$$ or $$2$$ and are $$\mathbb Q$$-Gorenstein if and only if $$H^0(S,K^{-m}_S)\not=0$$ for some $$m\geq 1$$. They are numerically Gorenstein if and only if $$H^0(S,K_S^{-1}\otimes L)\not=0$$ for some topologically trivial line bundle $$L$$ on $$S$$. A central part of the paper is devoted to the description of the discriminant of the quadratic form associated to a singularity. Let $$M(S)=(D_i.D_j)$$ be the self-intersection matrix of $$D$$. The singularities can be parametrized by the configuration of their dual graphs, represented by finite sequences $$\sigma$$ of integers.
The author introduces a family of polynomials $$P_\sigma$$ in $$N=N(\sigma)$$ variables, $$P_\sigma(\mathbb Z^N)\subset\mathbb Z$$, and numbers $$k_i=k_i(\sigma)\in\mathbb N$$, $$0\leq i\leq N-1$$. He obtains the existence of a sequence $$\sigma$$ with the property that $$\det M(S)=(-1)^n(P_\sigma(k_0,\dots,k_{N-1}))^2$$ and $$[H_2(S,\mathbb Z):\sum_{i=0}^{n-1}\mathbb ZD_i]=P_\sigma(k_0,\dots,k_{N-1})$$. The number $$\triangle_\sigma:=P_\sigma(k_0,\dots,k_{N-1})+1$$, the so-called twisting coefficient of the singularity, is a multiplicative invariant, i.e. $$\triangle_{\sigma\sigma'}=\triangle_\sigma\triangle_{\sigma'}$$, and equals the product of the determinants of the intersection matrices of the branches of the divisor $$D$$. The author points out the close connection of these invariants to global topological and analytical properties of surfaces with global spherical shells and to the classification of singular contracting germs of mappings and dynamical systems, see e.g. [G. Dloussky and K. Oeljeklaus, Ann. Inst. Fourier 49, No. 5, 1503–1545 (1999; Zbl 0978.32021)] and [Ch. Favre, J. Math. Pures Appl., IX. Sér. 79, No. 5, 475–514 (2000; Zbl 0983.32023)].

##### MSC:
 14E30 Minimal model program (Mori theory, extremal rays) 32J15 Compact complex surfaces 14E15 Global theory and resolution of singularities (algebro-geometric aspects) 14B05 Singularities in algebraic geometry 14E05 Rational and birational maps
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##### References:
 [1] Barth (W.), Hulek (K.), Peters (C.), and Van de Ven (A.).— Compact Complex Surfaces, Springer, Heidelberg, Second Edition (2004). · Zbl 1036.14016 [2] Demailly (J.P.).— Complex Analytic and Differential Geometry (1997). . [3] Dloussky (G.).— Structure des surfaces de Kato. Mémoire de la S.M.F $$112.\rm n^{∘ }14 (1984)$$. · Zbl 0543.32012 [4] Dloussky (G.).— Sur la classification des germes d’applications holomorphes contractantes. Math. Ann. 280, p. 649-661 (1988). · Zbl 0677.32004 [5] Dloussky (G.).— Une construction élémentaire des surfaces d’Inoue-Hirzebruch. Math. Ann. 280, p. 663-682 (1988). · Zbl 0617.14025 [6] Dloussky (G.).— On surfaces of class $$VII^+_0$$ with numerically anticanonical divisor, Am. J. Math. 128(3), p. 639-670 (2006). · Zbl 1102.32007 [7] Dloussky (G.), Oeljeklaus (K.).— Vector fields and foliations on compact surfaces of class $$VII_0$$. Ann. Inst. Fourier 49, p. 1503-1545 (1999). · Zbl 0978.32021 [8] Dloussky (G.), Oeljeklaus (K.).— Surfaces de la classe $$VII_0$$ et automorphismes de Hénon. C.R.A.S. 328, série I, p. 609-612 (1999). · Zbl 0945.32005 [9] Dloussky (G.), Oeljeklaus (K.), Toma (M.).— Class $$VII_0$$ surfaces with $$b_2$$ curves.Tohoku Math. J. 55, p. 283-309 (2003). · Zbl 1034.32012 [10] Enoki (I.).— Surfaces of class $$VII_0$$ with curves. Tôhoku Math. J. 33, p. 453-492 (1981). · Zbl 0476.14013 [11] Favre (Ch.).— Classification of $$2$$-dimensional contracting rigid germs, Jour. Math. Pures Appl. 79, p. 475-514 (2000). · Zbl 0983.32023 [12] Favre (Ch.).— Dynamique des applications rationnelles. Thèse pour le grade de Docteur en Sciences. Université de Paris XI Orsay (2000). [13] Hirzebruch (F.).— Hilbert modular surfaces. L’enseignement Math. 19, p. 183-281 (1973). · Zbl 0285.14007 [14] Inoue (M.).— New surfaces with no meromorphic functions II. Complex Analysis and Alg. Geom. p. 91-106. Iwanami Shoten Pb. (1977). · Zbl 0365.14011 [15] Karras (U.).— Deformations of cusps singularities. Proc. of Symp. in pure Math. 30, p. 37-44, AMS, Providence (1977). · Zbl 0352.14007 [16] Kato (Ma.).— Compact complex manifolds containing “global spherical shells” I Proc. of the Int. Symp. Alg. Geometry, Kyoto (1977) Iwanami Shoten Publ. · Zbl 0421.32010 [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). · Zbl 0137.17501 [18] Laufer (H.).— On minimally elliptic singularities. Amer. J. of Math. 99, p. 1257-1295, (1977). · Zbl 0384.32003 [19] Looijenga (E.) & Wahl (J.).— Quadratic functions and smoothing surface singularities. Topology 25, p. 261-291 (1986). · Zbl 0615.32014 [20] Mérindol (J.Y.).— Surfaces normales dont le faisceau dualisant est trivial. C.R.A.S. 293, p. 417-420 (1981). · Zbl 0482.14012 [21] Nakamura (I.).— On surfaces of class $$\rm VII_0$$ with curves. Proc. Japan Academy 58A, p. 380-383 (1982) · Zbl 0519.32017 [22] Nakamura (I.).— On surfaces of class $$\rm VII_0$$ with curves. Invent. Math. 78, p. 393-443 (1984). · Zbl 0575.14033 [23] Nakamura (I.).— On surfaces of class $$\rm VII_0$$ with Global Spherical Shells. Proc. of the Japan Acad. 59, Ser. A, No 2, p. 29-32 (1983). · Zbl 0536.14022 [24] Nakamura (I.).— On the equations $$x^p+y^q+z^r-xyz=0$$. Advanced Studies in pure Math. 8, Complex An. Singularities, p. 281-313 (1986). · Zbl 0643.14003 [25] Nakamura (I.).— Inoue-Hirzebruch surfaces and a duality of hyperbolic unimodular singularities I. Math. Ann. 252, p. 221-235 (1980). · Zbl 0425.14010 [26] Oeljeklaus (K.), Toma (M.).— Logarithmic moduli spaces for surfaces of class VII, Math. Ann. 341, p. 323-345 (2008). · Zbl 1144.32004 [27] Pinkham (H.).— Singularités rationnelles de surfaces. Appendice. Séminaire sur les singularités des surfaces. Lecture Notes 777. Springer-Verlag (1980). · Zbl 0459.14009 [28] Ribenboim (R.).— Polynomials whose values are powers. J. für die reine und ang. Math. 268/269, p. 34-40 (1974). · Zbl 0299.12103 [29] Riemenschneider (O.).— Familien komplexer Räume mit streng pseudokonvexer spezieller Faser. Comment. Math. Helvetici 39(51) p. 547-565 (1976). · Zbl 0338.32013 [30] Sakai (F.).— Enriques classification of normal Gorenstein surfaces. Am. J. of Math. 104, p. 1233-1241 (1981). · Zbl 0512.14022 [31] Serre (J.P.).— Cours d’arithmétique Presses Universitaires de France (1970). · Zbl 0376.12001 [32] Teleman (A.).— Projectively flat surfaces and Bogomolov’s theorem on class $$VII_0$$- surfaces, Int. J. Math., Vol.5, No 2, p. 253-264, (1994). · Zbl 0803.53038
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