The diversity of symplectic Calabi–Yau 6-manifolds

Given an integer b and a finitely presented group G, we produce a compact symplectic 6-manifold with c1 = 0, b2 > b, b3 > b and π1 = G. In the simply connected case, we can also arrange for b3 = 0; in particular, these examples are not diffeomorphic to Kähler manifolds with c1 = 0. The construction begins with a certain orientable, four-dimensional, hyperbolic orbifold assembled from right-angled 120-cells. The twistor space of the hyperbolic orbifold is a symplectic Calabi– Yau orbifold; a crepant resolution of this last orbifold produces a smooth symplectic manifold with the required properties.