Aspherical K\"ahler Manifolds with Solvable Fundamental Group

TL;DR

This paper reviews recent progress on the Benson-Gordon conjecture and classifies compact aspherical K"ahler manifolds with solvable fundamental groups, showing they are quotients of complex Euclidean space by discrete isometry groups.

Contribution

It proves the Albanese morphism is biholomorphic for certain K"ahler manifolds and classifies these manifolds up to biholomorphic equivalence.

Findings

Proof of the Benson-Gordon conjecture for K"ahler quotients of solvable Lie groups

Biholomorphic classification of compact aspherical K"ahler manifolds with solvable fundamental group

Identification of these manifolds as quotients of A2^n by discrete complex isometry groups

Abstract

We survey recent developments which led to the proof of the Benson-Gordon conjecture on K\"ahler quotients of solvable Lie groups. In addition we prove that the Albanese morphism of a K\"ahler manifold which is a homotopy torus is a biholomorphic map. The latter result then implies the classification of compact aspherical K\"ahler 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.