A six-dimensional compact symplectic solvmanifold without Kähler structures

Abstract

Gordon and Benson have proved that if a compact nilmanifold admits a Kahler structure then it is a torus [5] more precisely they proved that the condition (iv) fails for any symplectic structure on a non-toral nilmanifold M. This result was independently proved by Hasegawa [12] by showing that (v) fails for M. For a compact solvmanifold M of dimension 4 it is known that M has a Kahler structure if and only if it is a complex torus or a hyperelliptic surface. In fact, Auslander and Szczarba in [4] proved that if the first Betti number bι(M) of M is 2, M is a fiber bundle over T with fiber T. Then by Ue [19] M has a complex structure only if it is a hyperelliptic surface or a primary Kodaira surface which is a compact nilmanifold. Thus, if M is a Kahler manifold, it must be a hyperelliptic surface. Since !<&ι(M)<4, M can be a Kahler manifold only if it is a complex torus or a hyperelliptic surface. The fact that a hyperelliptic surface is a solvmanifold follows from Auslander [3]. The above result may be generalized as the following conjecture : A compact solvmanifold has a Kahler structure if and only if it is a finite quotient of a complex torus. In contrast to the case of compact nilmanifolds there are compact symplectic