Compact Kähler manifolds with nef anti-canonical bundle

In this paper, we prove that a compact Kähler manifold $X$ with the nef anti-canonical bundle $-K_X$ admits a locally trivial fibration $\phi \text{\hspace{0.17em}⁣}:X\to Y$, where the fiber $F$ is a rationally connected manifold and the base $Y$ is a Calabi–Yau manifold. We introduce a suitable approach that extends the strategy of Cao–Höring, originally developed for smooth projective varieties, to more general singular Kähler spaces. A key technical ingredient is a flatness criterion for pseudo-effective sheaves with vanishing first Chern class.