The diversity of symplectic Calabi-Yau six-manifolds

TL;DR

This paper constructs diverse examples of symplectic Calabi-Yau six-manifolds with prescribed topological properties, using hyperbolic orbifolds and twistor space techniques, expanding the known landscape of such manifolds.

Contribution

It introduces a method to produce symplectic Calabi-Yau six-manifolds with arbitrary fundamental groups and specified Betti numbers, not diffeomorphic to Kähler manifolds with c_1=0.

Findings

Constructed examples with arbitrary fundamental groups.

Produced simply-connected examples with b_3=0.

Demonstrated non-Kähler nature of the examples.

Abstract

Given an integer b and a finitely presented group G we produce a compact symplectic six-manifold with c_1 = 0, b_2 > b, b_3 > b and fundamental group G. In the simply-connected case we can also arrange for b_3 = 0; in particular these examples are not diffeomorphic to K\"ahler manifolds with c_1 = 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.