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**Geometric structures on compact complex analytic surfaces.**
*(English)*
Zbl 0602.57014

Starting from the recent determination of the Thurston geometries in dimension 4, this paper makes a study of manifolds with the various geometric structures. In preliminary sections, the list is recalled. There are 19 types of geometry (one of which is an infinite family), in 14 of which there is a complex structure (Kähler in 9 of the cases) compatible with a geometric structure. In each case the subgroups of the isometry group defining a geometry are listed; also general results about lattices in Lie groups are applied to give information about discrete cocompact subgroups.

Next the Enriques-Kodaira classification of compact complex surfaces is recalled. Within each class of surfaces there is a small list of possible geometric structures available, and in most cases one can give necessary and sufficient conditions for the complex surface to be geometric. The results are particularly neat for elliptic surfaces: such a surface is geometric if and only if all fibres are nonsingular and the base is a good orbifold, and the geometry is determined by the class. For any closed geometric 4-manifold M, the geometry is determined by the homotopy type of M - in fact the proof uses the homotopy invariants (signature, Euler numbers, fundamental group) exploited in the preceeding discussions.

Next the Enriques-Kodaira classification of compact complex surfaces is recalled. Within each class of surfaces there is a small list of possible geometric structures available, and in most cases one can give necessary and sufficient conditions for the complex surface to be geometric. The results are particularly neat for elliptic surfaces: such a surface is geometric if and only if all fibres are nonsingular and the base is a good orbifold, and the geometry is determined by the class. For any closed geometric 4-manifold M, the geometry is determined by the homotopy type of M - in fact the proof uses the homotopy invariants (signature, Euler numbers, fundamental group) exploited in the preceeding discussions.

### MSC:

57N13 | Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010) |

51M20 | Polyhedra and polytopes; regular figures, division of spaces |

32J15 | Compact complex surfaces |

32M10 | Homogeneous complex manifolds |

57R20 | Characteristic classes and numbers in differential topology |

57M05 | Fundamental group, presentations, free differential calculus |

22E40 | Discrete subgroups of Lie groups |