Numerical semiampleness and generalized minimal models

What OpenAI claims

Proves numerical semiampleness for nef adjoints K_X+B+M with K_X+B pseudo-effective and M nef rational, for projective klt rational pairs over algebraically closed characteristic-zero fields and smooth compact Kähler rational klt simple-normal-crossing pairs, using Bott–Chern cohomology in the latter case. Separately, projective generalized log canonical rational pairs over such fields admit minimal models for pseudo-effective adjoints and Mori fiber spaces otherwise, with nef b-data fixed.

Papers (4)

1. Numerical semiampleness of nef adjoint classes on compact Kähler manifolds

   6 October 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract 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.

2. Numerical Semiampleness of Nef Adjoint Divisors

   3 October 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract 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.

3. Minimal models and Mori fibre spaces for generalized log canonical Q-pairs

   24 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract 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.

4. Minimal models in numerical dimension one

   24 September 2026 · Preprint, not peer reviewed · PDF

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

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.