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Complex and Kähler structures on compact solvmanifolds. (English) Zbl 1120.53043
The author discusses recent results on the existence and classification problem of complex and Kähler structure on compact solvmanifolds. In particular, he determines all complex surfaces which are diffeomorphic to compact solvmanifolds. He proves the following theorem: A complex surface is diffeomorphic to a four-dimensional solvmanifold if and only if it is one of the following surfaces: complex tours, hyperelliptic surface, Inoue surface of type $$S^0$$, primary Kodaira surface, secondary Kodaira surface, Inoue surface of type $$S^{\pm}$$. Every complex structure on each of these complex surfaces is left-invariant.

##### MSC:
 53C55 Global differential geometry of Hermitian and Kählerian manifolds
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