This is a survey of the symplectic part of [7]. It is known that a hyperbolic manifold of even dimension is the base of a bundle whose total space admits a natural symplectic form ([11, 3]). We use this together with a construction resembling that of the Kummer surface to produce a simply-connected symplectic 6-manifold with vanishing first Chern class but no compatible complex structure. The role of the manifold-with-involution—the complex torus in Kummer’s original construction—is in our case played by a beautiful hyperbolic 4-manifold discovered by Davis [4].