Termination of projective and Kähler fourfold minimal model programs

Proves termination of every existing generalized log canonical flip sequence on globally Weil ℚ-factorial compact Kähler fourfolds, with rational boundary, fixed rational analytically nef b-data, and projective small flip diagrams with the prescribed ample signs. Also proves termination of arbitrary permitted minimal model programs for projective log canonical fourfolds with rational boundary in characteristic zero.

Manuscripts (5)

Termination of generalized log canonical flips on compact Kähler fourfolds

October 7, 2026 · source · cite as \cite{OAI:Termination-of-generalized-log-canonical-flips-on-compact-Kahler-fourfolds-October-7-2026}

Every sequence of generalized log canonical flips on a normal irreducible globally Weil ℚ-factorial compact Kähler fourfold terminates, provided the flips are projective small diagrams with the stated opposite ample signs. The boundary is rational, and the nef b-divisor is fixed and represented by an analytically nef ℚ-Cartier divisor on a projective modification. No scaling rule or pseudo-effectivity assumption is required. The theorem concerns existing flip sequences; it does not assert the existence of all contractions or flips.

Termination of generalized-canonical flips on compact Kähler fourfolds

October 6, 2026 · source · cite as \cite{OAI:Termination-of-generalized-canonical-flips-on-compact-Kahler-fourfolds-October-6-2026}

Every sequence of generalized-canonical flips on compact Kähler fourfolds with rational boundary and fixed rational analytically nef b-divisor terminates when the small morphisms are projective. We prove this for normal globally Weil ℚ-factorial models with boundary coefficients less than one, allowing exceptional log discrepancies equal to one. No scaling rule or pseudo-effectivity assumption is required. The theorem applies to existing projective small diagrams with the specified opposite ample signs.

Termination for projective log canonical fourfolds with rational boundary

September 24, 2026 · source · cite as \cite{OAI:Termination-for-projective-log-canonical-fourfolds-with-rational-boundary-September-24-2026}

We prove that every permitted minimal model program for a projective log canonical fourfold with rational boundary over an algebraically closed field of characteristic zero terminates. The result allows arbitrary negative extremal rays and mixed birational steps on the given models, without pseudo-effectivity or initial ℚ-factoriality.

Finite ordinary minimal model programs on compact Kähler fourfolds

October 6, 2026 · source · cite as \cite{OAI:Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds-October-6-2026}

We prove that every maximal ordinary negative-ray program starting from a compact Kähler klt fourfold pair with effective rational boundary in the global Weil-divisor ℚ-factorial category terminates. It ends at a nef model when the adjoint is pseudo-effective and at a projective Mori fibre space otherwise.

Finite ordinary minimal model programs on compact Kähler fourfolds

September 24, 2026 · source · cite as \cite{OAI:Finite-ordinary-minimal-model-programs-on-compact-Kahler-fourfolds-September-24-2026}

We prove that a globally Weil-ℚ-factorial compact Kähler klt fourfold pair with effective rational boundary and canonical rational line bundle admits a finite ordinary minimal model program starting on the given pair. It ends at a nef model when the adjoint class is pseudo-effective and at a Mori fibre space otherwise.

Result 056 · Algebraic and complex geometry. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.