Compact Kähler three-folds with nef anti-canonical bundle

In this paper, we prove that a non-projective compact Kähler three-fold with nef anti-canonical bundle is, up to a finite étale cover, one of the following: a manifold with vanishing first Chern class; the product of a K3 surface and the projective line; or a projective space bundle over a two-dimensional torus. This result extends Cao–Höring’s structure theorem for projective manifolds to compact Kähler manifolds in dimension 3. For the proof, we investigate the Minimal Model Program for compact Kähler three-folds with nef anti-canonical bundles by using the positivity of direct image sheaves, \(\mathbb {Q}\)-conic bundles, and orbifold vector bundles.