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On surfaces of class VII\(_ 0\) with curves. (English) Zbl 0575.14033
This paper is devoted to the characterization of compact \({\mathbb{C}}\)- analytic surfaces S which are: \[ (*)\text{ of class VII}_ 0,\quad b_ 1=1,\quad b_ 2>0\text{ and having at least one compact analytic curve.} \] It is known [K. Kodaira, Am. J. Math. 86, 751-798 (1964; Zbl 0137.175)] that those surfaces admit no non constant meromorphic functions and carry only finitely many compact connected analytic curves D; consequently, one always has \(D^ 2\leq 0.\)
First, the author exhibits 3 types of compact \({\mathbb{C}}\)-analytic surfaces satisfying (*): (1) Parabolic Inoue surfaces, \({\mathcal P}\), [which ar special cases of compact surfaces constructed by I. Enoki, Tôhoku Math. J., II. Ser. 33, 453-492 (1981; Zbl 0476.14013)] containing one elliptic curve and one cycle of rational curves. - (2) Hirzebruch-Inoue surfaces, \({\mathcal H}\), [M. Inoue, Complex Analysis Algebr. Geom., Collect. Pap. dedic. K. Kodaira, 91-106 (1977; Zbl 0365.14011)] containing 2 cycles of rational curves. - (3) Half-Inoue surfaces, \({\mathcal P}_{1/2}\) (M. Inoue, loc.cit.) containing a unique cycle of rational curves, C, with \(C^ 2<0\) and \(b_ 2(S)=\) number ofirreducible components of C.
A careful study of curves on those surfaces leads to the following main results: Theorem 1: Let S be a VII\(_ 0\) surface containing an elliptic curve and a cycle of rational curves. Then S is biholomorphic to some surface \({\mathcal P}\). - Theorem 2: Let S be a \(VII_ 0\) surface containing 2 cycles of rational curves. Then S is biholomorphic to some surface \({\mathcal H}\). - Theorem 3: Let S be a VII\(_ 0\) surface containing a unique cycle C of rational curves. Then S is biholomorphic to some \({\mathcal H}_{1/2}\) iff \(C^ 2=-b_ 2(S)\).
Reviewer: Vo Van Tan

MSC:
14J15 Moduli, classification: analytic theory; relations with modular forms
14J25 Special surfaces
32J25 Transcendental methods of algebraic geometry (complex-analytic aspects)
14J10 Families, moduli, classification: algebraic theory
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References:
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