Aspherical K��hler Manifolds with Solvable Fundamental Group

We survey recent developments which led to the proof of the Benson-Gordon conjecture on K��hler quotients of solvable Lie groups. In addition we prove that the Albanese morphism of a K��hler manifold which is a homotopy torus is a biholomorphic map. The latter result then implies the classification of compact aspherical K��hler manifolds with (virtually) solvable fundamental group up to biholomorphic equivalence. They are all biholomorphic to complex manifolds which are obtained as a quotient of $\bbC^n$ by a discrete group of complex isometries.