On compact Kähler manifolds with pseudo-effective tangent bundle

In this paper, we prove that a compact Kähler manifold $X$ with pseudo-effective (resp. singular positively curved) tangent bundle admits a smooth (resp. locally constant) rationally connected fibration $\phi \text{\hspace{0.17em}⁣}:X\to Y$ onto a finite étale quotient $Y$ of a compact complex torus. This result extends the structure theorem previously established for smooth projective varieties to compact Kähler manifolds.