# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] Artin, M.: Some numerical criteria for contractibility of curves on algebraic surfaces. Amer. J. Math.84, 485-496 (1964) · Zbl 0105.14404 [2] Enoki, I.: Surfaces of class VII0 with curves. Tohoku Math. J.33, 453-492 (1981) · Zbl 0476.14013 [3] Grauert, H.: Über Modifikationen und exzeptionelle analytische, Mengen. Math. Ann.146, 331-368 (1962) · Zbl 0173.33004 [4] Inoue, M.: On surfaces of class VII0. Invent. Math.24, 269-310 (1974) · Zbl 0283.32019 [5] Inoue, M.: New surfaces with no meromorphic functions. Proc. Int. Cong. Math., Vancouver, vol. 1, 423-426 (1974) [6] Inoue, M.: New surfaces with no meromorphic functions. II. Complex Analysis and Algebraic Geometry, Iwanami Shoten Publ. and Cambridge Univ. Press, pp. 91-106 (1977) [7] Karras, U.: Deformations of cusp singularities. Proc. of Symposia in Pure Math.30, 37-44 (1977) · Zbl 0352.14007 [8] Kato, Ma.: Compact complex manifolds containing ?global spherical shells?. I. Proc. Int. Symp. Algebraic Geometry, Kyoto, pp. 45-84 (1977) · Zbl 0421.32010 [9] Kato, Ma.: On a certain class of non-algebraic non-Kähler compact complex manifolds. Preprint · Zbl 0514.32017 [10] Kawamata, Y.: On deformations of compactifiable complex manifolds. Math. Ann.235, 247-265 (1978) · Zbl 0371.32017 [11] Kodaira, K.: On compact complex analytic surface, I, II. Ann. Math.71, 111-152 (1960);77, 563-626 (1962) · Zbl 0098.13004 [12] Kodaira, K.: On the structure of compact complex analytic surfaces, I, II, III, IV. Amer. J. Math.,86, 751-798 (1964);88, 682-721 (1966);90, 55-83 (1968);90, 170-192 (1968) · Zbl 0137.17501 [13] Kodaira, K.: On stability of compact submanifolds of complex manifolds. Amer. J. Math.85, 79-94 (1963) · Zbl 0173.33101 [14] Laufer, H.: Normal two dimensional singularities. Annals of Math. Studies, Princeton Univ. Press, (1971) · Zbl 0245.32005 [15] Laufer, H.: Taut two dimensional singularities. Math. Ann.205, 131-164 (1973) · Zbl 0281.32010 [16] Laufer, H.: Versal deformation for two dimensional pseudoconvex manifolds. Annali d. Scuola Nurm. Sup. di Pisa7, 511-521 (1980) · Zbl 0512.32016 [17] Nakamura, I.: Inoue-Hirzebruch surfaces and a duality of hyperbolic unimodular singularities, I. Math. Ann.252, 221-235 (1980) · Zbl 0436.14010 [18] Nakamura, I.: On surfaces of class VII0 with curves. Proc. Japan. Acad.58A, 380-383 (1982) · Zbl 0519.32017 [19] Nakamura, I.: On surfaces of class VII0 with global spherical shells. Proc. Japan Acad.58A, 29-32 (1983) · Zbl 0536.14022 [20] Nakamura, I.: VII0 surfaces and a duality of cusp singularities,Classification of algebraic and analytic manifolds, Progress in Mathematics, vol. 39, pp. 333-378, Birkhäuser 1983 [21] Oda, T.: Torus embeddings and applications. Tata Inst. Lecture Notes, Bombay (1978) · Zbl 0417.14043
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.