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Termination of projective and Kähler fourfold minimal model programs

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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.

The release has no Lean statement for this family, so there is no formal statement to compare with the result described above.

Five current papers; three replaced by later editions.

The paper states

Theorem 1.1

In Termination of generalized log canonical flips on compact Kähler fourfolds, 7 October 2026

Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.

Four other possible matches are listed below the card, among them Theorem 1.1 in Termination of generalized-canonical flips on compact Kähler fourfolds and Theorem 1.2 (Termination of ordinary programs) in Finite ordinary minimal model programs on compact Kähler fourfolds.

Let X_0 be a normal irreducible globally Weil \Q-factorial compact Kähler fourfold. Let B_0 be an effective rational Weil divisor of finite support with coefficients in [0,1]. Let \M be represented by an analytically nef \Q-Cartier divisor on a projective bimeromorphic normal compact Kähler model of X_0. Fix compatible actual canonical divisor representatives on all birational models under consideration. Suppose that D_0=K_{X_0}+B_0+M_{X_0} is \Q-Cartier and that (X_0,B_0+\M) is generalized log canonical.

Consider a sequence of normal irreducible globally Weil \Q-factorial compact Kähler fourfolds

X_0\dashrightarrow X_1\dashrightarrow X_2\dashrightarrow\cdots.

On X_i, let B_i be the strict transform of B_0, let M_i=M_{X_i} be the trace of the same b-divisor, and assume that D_i=K_{X_i}+B_i+M_i is \Q-Cartier. Transport the canonical representatives under the induced identifications of prime divisors. Suppose that every step is given by a diagram

A diagram here is drawn in the paper: see it at the passage.

where Z_i is normal compact Kähler, both maps are projective small bimeromorphic morphisms with connected fibers, neither is an isomorphism, and

-D_i\text{ is }f_i\text{-ample},\qquad D_{i+1}\text{ is }f_i^+\text{-ample}.

Here small means that the exceptional locus contains no prime divisor. Then the sequence is finite.

Four more possible matches

  • Theorem 1.1 in Termination of generalized-canonical flips on compact Kähler fourfolds, 6 October 2026.
  • Theorem 1.2 (Termination of ordinary programs) in Finite ordinary minimal model programs on compact Kähler fourfolds, 6 October 2026.
  • Theorem 1.2 (Finite ordinary programs) in Finite ordinary minimal model programs on compact Kähler fourfolds, 24 September 2026.
  • Theorem 1.2 in Termination for projective log canonical fourfolds with rational boundary, 24 September 2026.
Reported proof verification
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Papers and explanations
Five current papers, each readable here as a page beside its PDF.

Papers

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Termination of generalized log canonical flips on compact Kähler fourfolds

7 October 2026, current edition.

Read it herePDF

This version clarifies the independent relative MMP and birational-comparison inputs used by the proof and updates citations to revised companion papers; see the previous version.

Earlier editions: 5 October 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

6 October 2026, current edition.

Read it herePDF

This version clarifies that only independent negativity, codimension-two smoothness, and cycle-comparison results are used from the companion fourfold paper; see the previous version.

Earlier editions: 5 October 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

24 September 2026, current edition.

Read it herePDF

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

6 October 2026, current edition.

Read it herePDF

This version adds carrier-positivity and multiplier-ideal arguments and revises the contraction induction for ordinary negative-ray programs; see the previous version.

Earlier editions: 5 October 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

24 September 2026, current edition.

Read it herePDF

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.

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  • An unsigned public tier list, the OpenAI Math Tier List, classes it as a proof and rates its significance, if it holds up, as “Major math journal”. The page says that nothing on it is verified.

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