Irreducible 4-manifolds need not be complex. (English) Zbl 0805.57012

The authors construct smooth 4-manifolds which are homeomorphic to the Kummer surface but admit no structure of a complex or algebraic surface. This gives the first counterexamples to the so-called decomposition conjecture which states that every smooth closed simply-connected 4- manifold is diffeomorphic to a connected sum of algebraic surfaces. To construct their examples, the authors start with an elliptic structure of the Kummer surface and apply self-diffeomorphisms to it to obtain further (nonequivalent) elliptic structures. Then they perform logarithmic transformations (viewed as the differential topology operation of cutting out and gluing back in neighborhoods of generic fibers) simultaneously to generic fibers from different elliptic fibrations which do not share any holomorphic structure. The resulting 4-manifolds turn out to be noncomplex by Kodaira’s classification and computations of Donaldson invariants for elliptic surfaces and the 4-manifolds obtained by the above cut and glue operations.
Reviewer: P.Teichner (Mainz)


57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
57R57 Applications of global analysis to structures on manifolds
14J25 Special surfaces
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