On the classification of smooth compact \(\mathbb{C}^*\)-surfaces (Q1891223)

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.