Aspherical Kähler manifolds with solvable fundamental group

Abstract

Abstract. We survey recent developments which led to the proof of the Benson-Gordon conjecture on Kähler quotients of solvable Lie groups. In addition we prove that the Albanese morphism of a Kähler manifold which is a homotopy torus is a biholomorphic map. The latter result then implies the classification of compact aspherical Kähler manifolds with (virtually) solvable fundamental group up to biholomorphic equivalence. They are all biholomorphic to complex manifolds which are obtained as a quotient of C n by a discrete group of complex isometries.The purpose of this article is to give an account on recent developments concerning the classification of compact aspherical Kähler manifolds whose fundamental groups contain a solvable subgroup of finite index. This problem stirred some interest among differential geometers for quite some time. It traces back to the classification of Kähler Lie groups and became popular through a conjecture of Benson and Gordon on Kähler quotients of solvable Lie groups. Only recently satisfying answers to the original Benson-Gordon conjecture became available when methods from complex geometry were introduced into the subject. The story as it presents itself now may be seen as an interesting show case where research questions from differential geometry, complex geometry and topology meet.Outline of the paper. The first section of the paper has a survey character. We introduce the Benson-Gordon conjecture and describe its relation to the classification of Kähler Lie groups, Kähler groups and Kähler manifolds. In the course we also sketch the proof of a theorem of Arapura and Nori on polycyclic Kähler groups. This leads us to the following rigidity result for Kähler infra-solvmanifolds, which, in particular, contains the solution to the conjecture of Benson and Gordon.Theorem A. Let M be an infra-solvmanifold which admits a Kähler metric. Then M is diffeomorphic to a flat Riemannian manifold.In section 2 we develop generalisations of the previously described results. These concern the classification of Kähler structures on general smooth aspherical manifolds with polycyclic fundamental group. One new key point is to establish that a

Virtually abelian Kähler and projective groups

“…For this, recall that a two-form ω ∈ 2 (R n ) * is called a polarisation for J (with respect to the lattice Z n ) if the following conditions are satisfied: 1. ω(J ·, J ·) = ω(·, ·), 2. g = ω(J ·, ·) is positive definite, 3. ω(Z n , Z n ) ⊆ Z.…”

Some results on cosymplectic manifolds

“…Many results are known about Kähler Lie groups (see for instance [3]). Basic examples are flat Kähler Lie groups and solvable Kähler Lie groups acting simply transitively on a bounded domain in C n by biholomorphisms (see [17]).…”

Exotic smooth structures and symplectic forms on closed manifolds

“…In the present paper we will give a short proof of the fact that there are no Kähler structures on exotic tori. When this paper was written we have learned that this fact was known to algebraic geometers [BC,C1].…”

“…As a result of efforts of [A,AN,Ha,Ha1,TK] together with developments from algebraic geometry (see [BC,C1]) the final (affirmative) solution of this problem was recently achieved (See Section 3). To the authors knowledge, the final clean proof was given in [Ha,Ha1].…”

“…Thus this paper can be considered as complementary to [BC,C1], with the intersection exactly in the original Benson-Gordon problem.…”