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Math-2545 · mathematics · Algebraic geometry

Generalized-canonical flip termination on Kähler fourfolds

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Fix a normal globally Weil-Q-factorial compact Kähler fourfold, rational boundary coefficients in [0,1), and nef rational b-Cartier data on a projective birational Kähler model. Assume generalized discrepancies are positive for all primes and at least one for exceptional primes. Must every sequence of nontrivial projective small rational-adjoint flips, preserving these canonical and b-data, be finite?

Short English statement

Fix a normal globally Weil-Q-factorial compact Kähler fourfold, rational boundary coefficients in [0,1), and nef rational b-Cartier data on a projective birational Kähler model. Assume generalized discrepancies are positive for all primes and at least one for exceptional primes. Must every sequence of nontrivial projective small rational-adjoint flips, preserving these canonical and b-data, be finite?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2545-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2545

Manuscript 1 (family 056, 2026-10-05): The problem selects the mathematical statement summarized here, including its stated hypotheses and parameter range. The cited theorem gives a claimed proof of that assertion in full.

Generalized-canonical flip termination on Kähler fourfolds

OpenAI's claim

Generalized-canonical flip sequences with fixed rational nef data on compact Kähler fourfolds are claimed to terminate even when exceptional discrepancies equal one. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
Start with a normal irreducible globally Weil rational-factorial compact Kähler fourfold $X_0$, effective rational boundary $B_0$ of coefficients below one, and a fixed rational nef b-divisor $\mathbf M$ represented on a projective birational compact Kähler model. Require generalized log discrepancies positive for every prime and at least one for exceptional primes. For strict transforms $B_i$, traces $M_i$ and rational Cartier $D_i=K_{X_i}+B_i+M_i$, OpenAI claims every sequence of projective small nonisomorphic bimeromorphic diagrams with connected fibers \[ X_i\xrightarrow{f_i}Z_i\xleftarrow{f_i^+}X_{i+1},\qquad -D_i\text{ relatively ample for }f_i,\quad D_{i+1}\text{ relatively ample for }f_i^+, \] is finite, provided every $Z_i$ is normal compact Kähler, all models are normal globally Weil rational-factorial compact Kähler fourfolds, and canonical representatives are compatible. Small means no exceptional prime divisor. This theorem has a generalized-canonical discrepancy hypothesis, not unrestricted generalized log canonicity.

Outline of the argument

Manuscript 1 (family 056, 2026-10-05): A corrected finite count of low-discrepancy places first gives ordinary terminality along general points of flipped surfaces. Fixed denominator gaps and analytic surface-cycle comparisons then force a finite decrease, including zero exceptional boundary coefficients and nonnormal boundary components.

Formal verification

Family 056 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.

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Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

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