Numerical semiampleness and generalized minimal models

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 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.