1 · The essentials
These manuscripts study whether a process for changing a geometric space into a simpler form must eventually stop. The spaces considered have four dimensions in the mathematical sense. The main manuscript shows that certain sequences of changes called flips cannot continue forever for the compact complex spaces it considers. This conclusion requires specific conditions on the spaces, on extra data attached to them, and on each allowed change. It concerns sequences whose steps already exist; it does not establish that every desired step can be carried out. A companion manuscript shows termination for every permitted simplification program in a related setting of projective spaces. The main result is not formalized in Lean and has not been formally verified.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine rearranging a room under strict rules about which changes are allowed. A termination statement says that any sequence obeying those rules must stop, rather than allowing endless rearrangement. It does not, by itself, guarantee that a permitted next move is available.
Why it matters
Termination distinguishes a finite geometric procedure from one that might continue indefinitely. These manuscripts address termination without requiring a scaling rule or a pseudo-effectivity assumption in the stated flip results. The ordinary Kähler programs described in the companion manuscripts also identify the possible endpoints: a nef model or a Mori fibre space, according to the adjoint's pseudo-effectivity.
2 · Going further
The main problem is termination of generalized log canonical flips on a normal, irreducible, globally Weil -factorial compact Kähler fourfold. The boundary is rational, and the nef b-divisor is fixed, represented by an analytically nef -Cartier divisor on a projective modification. Every flip must be a projective small diagram with the specified opposite ample signs. Under these hypotheses, termination of every existing flip sequence is shown in the manuscript, without a scaling rule or a pseudo-effectivity assumption.
The supplied abstracts state the results and their scope, but do not describe the argument establishing termination. In particular, they provide no decreasing quantity or other mechanism from which to explain why infinitely many flips are excluded. The generalized-canonical companion treats normal globally Weil -factorial compact Kähler models with rational boundary coefficients less than one, fixed rational analytically nef b-data, and projective small morphisms with the specified ample signs. It allows exceptional log discrepancies equal to one and likewise requires neither scaling nor pseudo-effectivity.
The projective companion concerns log canonical fourfolds with rational boundary over an algebraically closed field of characteristic zero. It shows termination of every permitted minimal model program, allowing arbitrary negative extremal rays and mixed birational steps on the given models. Neither pseudo-effectivity nor initial -factoriality is assumed there; this is a different scope from the main Kähler flip statement.
The two ordinary Kähler companions concern globally Weil -factorial klt fourfold pairs with effective rational boundary. Manuscript 4 shows that every maximal ordinary negative-ray program terminates, ending at a nef model if the adjoint is pseudo-effective and at a projective Mori fibre space otherwise. Manuscript 5 additionally assumes a canonical rational line bundle and shows that a finite ordinary minimal model program exists starting on the given pair, with a nef model or Mori fibre space as its endpoint according to the same distinction.
Useful notions
- Complex and projective geometric spaces
- Divisors and rational boundaries
- Birational maps and flips
- Singularity conditions for pairs
- Nefness and pseudo-effectivity
What was known before / what is new
Before
The source presents these results in the language of flips and minimal model programs in projective and compact Kähler geometry. It does not describe earlier termination results, attribute an open problem, or supply a historical baseline against which to measure the claims.
New
The main manuscript shows termination for every existing generalized log canonical flip sequence under its stated compact Kähler fourfold hypotheses. The projective companion shows termination for every permitted log canonical fourfold program with rational boundary in characteristic zero, including arbitrary negative extremal rays and mixed birational steps. The remaining companions address generalized-canonical flips and finite ordinary Kähler programs, with their distinct assumptions and endpoint statements.
Limits and assumptions
The main result is restricted to normal irreducible globally Weil -factorial compact Kähler fourfolds, rational boundary, fixed rational analytically nef b-data represented on a projective modification, and projective small flip diagrams with the prescribed opposite ample signs. It addresses existing sequences, not the existence of all contractions or flips. The projective result concerns permitted programs over an algebraically closed field of characteristic zero; the ordinary Kähler statements require klt pairs and effective rational boundary, with an additional canonical rational line bundle hypothesis in Manuscript 5. The abstracts do not supply an argument or specify which questions beyond these scopes remain open. The main result is not formalized in Lean and has not been formally verified.
Mini glossary
- Fourfold —
- A geometric space of dimension four.
- Flip —
- A birational change appearing in the sequences studied here, required in the Kähler statements to fit a projective small diagram with specified opposite ample signs.
- Termination —
- The assertion that a sequence of allowed steps cannot continue indefinitely.
- Rational boundary —
- The boundary divisor attached to the space, with rational coefficients.
- Minimal model program —
- A program of permitted geometric steps whose termination and, in the ordinary Kähler statements, endpoints are studied in these manuscripts.
Keywords : Termination · Fourfolds · Minimal model programs
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
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.
To be verified: the proof is written in the manuscript, but no software has checked it and no validation by independent mathematicians is established. It may contain errors.
Manuscripts in this family
Termination of generalized log canonical flips on compact Kähler fourfolds
07/10/2026Every 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
06/10/2026Every 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
24/09/2026We 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
06/10/2026We 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
24/09/2026We 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.