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.


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
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