Rational curves on compact K\"ahler manifolds

We present an inductive strategy to show the existence of rational curves on compact Kaehler manifolds which are not minimal models but have a pseudoeffective canonical bundle. The tool for this inductive strategy is a weak subadjunction formula for lc centers associated to certain big cohomology classes. This subadjunction formula is based, as in the projective case, on positivity arguments for relative adjoint classes.