On compact K\"ahler manifolds with pseudo-effective tangent bundle

In this paper, we prove that a compact K\"ahler manifold $X$ with pseudo-effective (resp. singular positively curved) tangent bundle admits a smooth (resp. locally constant) rationally connected fibration $\phi \colon X \to Y$ onto a finite \'etale quotient $Y$ of a compact complex torus. This result extends the structure theorem previously established for smooth projective varieties to compact K\"ahler manifolds.