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.


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
Full Text: DOI