Let X be a smooth connected compact Kähler manifold, let B be an effective rational simple normal crossing divisor with coefficients less than one, and let M be a nef rational holomorphic line bundle on X. If $K_X+B$ is pseudo-effective and $K_X+B+M$ is nef, we prove that its first Chern class in real Bott–Chern cohomology is represented by a semiample rational line bundle. This numerical statement allows a flat change of line bundle; it does not assert semiampleness of the original adjoint.

We prove the Generalised Abundance Conjecture: if $(X,B)$ is a projective klt ℚ-pair over an algebraically closed field of characteristic zero, $K_X+B$ is pseudo-effective, M is a nef ℚ-Cartier divisor on X, and $K_X+B+M$ is nef, then $K_X+B+M$ is numerically equivalent to a semiample ℚ-Cartier divisor on X.

We resolve the existence form of the minimal-model conjecture for projective generalized log canonical ℚ-pairs over algebraically closed fields of characteristic zero. Such a pair admits a minimal model when its adjoint divisor is pseudo-effective, and a Mori fibre space otherwise, while keeping the nef ℚ-Cartier b-divisor fixed. Taking the nef part to be zero gives the corresponding result for ordinary log canonical ℚ-pairs.

We resolve the numerical-dimension-one case of the minimal-model conjecture for smooth connected complex projective varieties of dimension at least three. If K_X is pseudo-effective and $\kappa_\sigma(X,K_X)=1$, with κ_σ defined by section growth with a fixed ample twist, then X admits a projective ℚ-factorial terminal minimal model.