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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)
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