## Structure des surfaces de Kato.(English)Zbl 0543.32012

M. Kato first studied complex surfaces containing ”global spherical shells” [in Proc. Int. Symp. Algebraic Geometry, Kyoto 1977, 45-84 (1977; Zbl 0421.32010)] and proved that the Inoue surfaces contain global spherical shells. In this paper such minimal surfaces are studied in general, and when $$b_ 2(S)>0$$ they are called Kato surfaces (when $$b_ 2(S)=0$$ it is the case of primary Hopf surfaces which have been already thoroughly studied). In the first part, different kinds of invariants are attached to Kato surfaces: germs of contracting mappings between surfaces, a complex number ”the trace”, the family of integers: the family of the opposite of self-intersection of rational curses of the universal covering space. All possible curves are given. - In the second part, Kato surfaces with ”non vanishing trace” are studied by means of germs of mappings. A formal curve and formal vector fields invariant by these germs are introduced and it can be checked on examples that these objects are not convergent in general. All intersection matrices of rational curves are described and it is shown that when ”trace” vanishes the matrix is negative-definite.

### MSC:

 32J15 Compact complex surfaces 14J15 Moduli, classification: analytic theory; relations with modular forms 53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature 32J25 Transcendental methods of algebraic geometry (complex-analytic aspects)

Zbl 0421.32010
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### References:

 [1] F.A. BOGOMOLOV : Surfaces of class VII and affine geometry (Russian) , Izv. Akad. Nauk SSSR ser. Mat. 46 ( 1982 ) pp. 710-761. MR 84k:32034 | Zbl 0527.14029 · Zbl 0527.14029 [2] G. DLOUSSKY : Sur les surfaces compactes de la classe VII contenant une coquille sphérique globale . CRAS, T. 292 pp. 727-730 ( 1981 ). MR 82f:32042 | Zbl 0472.32019 · Zbl 0472.32019 [3] G. DLOUSSKY : Sur les courbes et champs de vecteurs globaux des surfaces analytiques de la classe VII admettant une coquille sphérique globale . CRAS t. 295 pp. 111-114 ( 1982 ). MR 83j:32031 | Zbl 0505.32022 · Zbl 0505.32022 [4] I. ENOKI : Surfaces of class VII with curves . Proceedings Japan Acad. 56 ser. A ( 1980 ). Article | MR 82d:32022 | Zbl 0462.32012 · Zbl 0462.32012 [5] I. ENOKI : Surfaces of class VII with curves . Tôhoku Math. J. 33 ( 1981 ) 453-492. Article | MR 83g:32028 | Zbl 0476.14013 · Zbl 0476.14013 [6] H. FREUDENTHAL : Über die Enden topologischer Räume und Gruppen . Math. Zeitschrift 33 ( 1931 ) 692-713. Zbl 0002.05603 | JFM 57.0731.01 · Zbl 0002.05603 [7] H. GRAUERT : Über Modifikationen und exzeptionnelle analytischen Mengen . Math. Annalen 146 ( 1962 ), 331-368. MR 25 #583 | Zbl 0178.42702 · Zbl 0178.42702 [8] C. HOUZEL : Séminaire Cartan 1960 - 1961 Exposé 21 E.N.S. Paris. Numdam [9] M. INOUE : On surfaces of class VII . Invent. Math. 24 ( 1974 ) pp. 269-310. MR 49 #7479 | Zbl 0283.32019 · Zbl 0283.32019 [10] M. INOUE : New surfaces with no meromorphic function . Proceedings of the Int. Congress of Math. Vancouver 1974 Vol 1 R. James Ed. 1975 pp. 423-426. MR 56 #682 | Zbl 0365.14010 · Zbl 0365.14010 [11] M. INOUE : New surfaces with no meromorphic functions II Complex Analysis and Algebraic Geometry . Iwanami Shoten Publ. Tokyo 1977 . Zbl 0365.14011 · Zbl 0365.14011 [12] Ma. KATO : Compact complex manifolds containing ”global spherical shells” I Proceedings of the Int. Symp. Algebraic Geometry, Kyoto 1977 . Kinokuniya Book Store. Tokyo 1978 . Zbl 0421.32010 · Zbl 0421.32010 [13] K. KODAIRA : On compact complex analytic surfaces I, II, III . Annals of Math. vol. 71 ( 1960 ) pp. 111-152 ; vol. 77 ( 1963 ) pp. 563-626 ; vol. 78 ( 1963 ) pp. 1-40. Zbl 0098.13004 · Zbl 0098.13004 [14] K. KODAIRA : On the structure of compact complex analytic surfaces I, II . Am. J. of Math. vol 86 ( 1964 ) pp. 751-798 ; vol. 88 ( 1966 ) pp. 682-721. Zbl 0137.17501 · Zbl 0137.17501 [15] H. LAUFER : Normal two-dimensional singularities . Annals of Math. Studies Princetion Univ. Press ( 1971 ). MR 47 #8904 | Zbl 0245.32005 · Zbl 0245.32005 [16] I. NAKAMURA : On surfaces of class VII with positive b2 . Preprint. [17] I. NAKAMURA : On surfaces of class VII with curves . Proceedings of the Japan Acad. vol. 58 ( 1982 ), sér. A, pp. 380-383. Article | MR 84e:32027 | Zbl 0519.32017 · Zbl 0519.32017 [18] I. NAKAMURA : On surfaces of class VII with global spherical shells . Proceedings of the Japan Acad., vol. 59, ser. A, n^\circ 2 ( 1983 ) pp. 29-32. Article | MR 85a:32034 | Zbl 0536.14022 · Zbl 0536.14022 [19] I. NAKAMURA : On surfaces of class VII with curves . Preprint. · Zbl 0575.14033 [20] H. ROSSI : Vector fields on analytic spaces . Annals of Math. vol. 78 ( 1963 ) 455-467. MR 29 #277 | Zbl 0129.29701 · Zbl 0129.29701 [21] W. RUDIN : Function theory in the unit ball of \Bbb Cn . Springer Verlag 1980 . MR 82i:32002 | Zbl 0495.32001 · Zbl 0495.32001
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