×

zbMATH — the first resource for mathematics

On the classification of smooth compact \(\mathbb{C}^*\)-surfaces. (Zur Klassifikation glatter kompakter \(\mathbb{C}^*\)-Flächen.) (German) Zbl 0830.32012
The author classifies smooth compact complex surfaces with a nontrivial holomorphic \(\mathbb{C}^*\)-action possessing fixed points. He reduces the problem to minimal surfaces and proves the main result: The minimal surfaces as above belong to three classes: (i) algebraic surfaces, (ii) Hopf surfaces, (iii) parabolic Inoue surfaces. (The \(\mathbb{C}^*\)-actions are described by the author). He also gives the list of all the minimal compact almost homogeneous complex surfaces. These are: (i) minimal rational surfaces, (ii) topologically trivial \(\mathbb{P}_1\)-bundles over a one-dimensional complex torus, (iii) Hopf surfaces with an abelian fundamental group, (iv) two-dimensional complex tori.

MSC:
32J15 Compact complex surfaces
32M05 Complex Lie groups, group actions on complex spaces
32M12 Almost homogeneous manifolds and spaces
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] W. Barth, C. Peters, A. Van de Ven: Compact Complex Surfaces. Berlin, Heidelberg, New York, Tokyo: Springer 1984
[2] A. Blanchard: Un théorème sur les automorphismes d’une variété algébrique projective, Application aux fibrées et à une conjecture d’ Igusa. C. R. Acad. Sci. Paris.240, 2198-2201 (1955) · Zbl 0065.38804
[3] G. E. Bredon: Introduction to compact transformation groups. New York: Academic Press 1972 · Zbl 0246.57017
[4] J. Carrell, A. Howard, C. Kosniowski: Holomorphic Vector Fields on Complex Surfaces. Math. Ann.204, 73-81 (1973) · Zbl 0253.14005 · doi:10.1007/BF01431490
[5] G. Dloussky: Sur les courbes et champs de vecteurs globaux des surfaces analytiques de la classe VII0 admettant une coquille sphérique globale. C. R. Acad. Sci. Paris,295 (1982), 111-114. · Zbl 0505.32022
[6] I. Enoki: Surfaces of Class VII0 with Curves. Tohoku Math. Journ.33, 453-492, (1981) · Zbl 0476.14013 · doi:10.2748/tmj/1178229349
[7] Ch. Gellhaus, P. Heinzner: Komplexe Flächen mit holomorphen Vektorfeldern. Abh. Math. Sem. Univ. Hamburg.60, 37-46 (1990) · Zbl 0734.32017 · doi:10.1007/BF02941046
[8] P. Heinzner: Fixpunktmengen kompakter Gruppen in Teilgebieten Steinscher Mannigfaltigkeiten. J. reine angew. Math.402, 128-137 (1989) · Zbl 0673.32026 · doi:10.1515/crll.1989.402.128
[9] P. Heinzner: Geometric invariant theory on Stein spaces. Math. Ann.289, 631-662 (1991) · Zbl 0728.32010 · doi:10.1007/BF01446594
[10] M. Kato: Compact complex manifolds containing ?global spherical shells?, I. Intl. Symp. on Algebraic Geometry, Kyoto 1977, (M. Nagata, ed.) Kinokuniya Book-Store, Tokyo, 45-84 (1978)
[11] K. Kodaira: On the structure of compact complex analytic surfaces II. Amer. J. of Math.88, 682-721 (1966) · Zbl 0193.37701 · doi:10.2307/2373150
[12] I. Nakamura: On surfaces of class VII0 with curves. Invent Math.78, 393-443 (1984) · Zbl 0575.14033 · doi:10.1007/BF01388444
[13] P. Orlik, P. Wagreich: Algebraic surfaces withk *-actions. Acta Math.138, 43-81 (1977) · Zbl 0352.14016 · doi:10.1007/BF02392313
[14] J. Potters: On Almost Homogeneous Compact Complex Surfaces. Inv. math.8, 244-266 (1969) · Zbl 0205.25102 · doi:10.1007/BF01406077
[15] R. Remmert, A. Van de Ven: Zur Funktionentheorie homogener komplexer Mannigfaltigkeiten. Topology2, 137-157 (1963) · Zbl 0122.08602 · doi:10.1016/0040-9383(63)90029-6
[16] A. Van de Ven: Analytic compactification of complex homology cells. Math. Ann.147, 189-204 (1962) · Zbl 0105.14406 · doi:10.1007/BF01470739
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.