Compact K\"ahler three-folds with nef anti-canonical bundle

In this paper, we prove that a non-projective compact K\"ahler three-fold with nef anti-canonical bundle is, up to a finite \'etale 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 $2$-dimensional torus. This result extends Cao-H\"oring's structure theorem for projective manifolds to compact K\"ahler manifolds in dimension $3$. For the proof, we investigate the Minimal Model Program for compact K\"ahler three-folds with nef anti-canonical bundles by using the positivity of direct image sheaves, $\mathbb{Q}$-conic bundles, and orbifold vector bundles.