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Math-2547 · mathematics · Algebraic geometry

Generalized lc flip termination on Kähler fourfolds

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
On globally Weil-Q-factorial normal compact Kähler fourfolds, fix effective rational boundary and a nef rational b-Cartier divisor represented on a projective birational Kähler model, with the generalized pair lc. Must every sequence of existing projective small connected-fiber flips for this same rational adjoint and b-data terminate? Each flipping adjoint is negative ample, and its next transform positive ample.

Short English statement

On globally Weil-Q-factorial normal compact Kähler fourfolds, fix effective rational boundary and a nef rational b-Cartier divisor represented on a projective birational Kähler model, with the generalized pair lc. Must every sequence of existing projective small connected-fiber flips for this same rational adjoint and b-data terminate? Each flipping adjoint is negative ample, and its next transform positive ample.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2547-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2547

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 lc flip termination on Kähler fourfolds

OpenAI's claim

Every existing generalized log-canonical flip sequence in the specified globally factorial compact Kähler fourfold category is claimed to terminate, without scaling or pseudo-effectivity assumptions. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
Let $X_0$ be normal irreducible globally Weil rational-factorial compact Kähler of dimension four, with effective rational finite-support $B_0$ of coefficients in $[0,1]$ and a fixed analytically nef rational b-divisor represented on a projective bimeromorphic compact Kähler model. Suppose $D_0=K_{X_0}+B_0+M_0$ rational Cartier and all generalized log discrepancies nonnegative. With strict transforms, the same b-divisor and compatible canonical representatives, OpenAI claims finiteness of every sequence \[ X_i\xrightarrow{f_i}Z_i\xleftarrow{f_i^+}X_{i+1},\qquad -D_i\text{ }f_i\text{-ample},\quad D_{i+1}\text{ }f_i^+\text{-ample}, \] with every $Z_i$ normal compact Kähler, whose maps are projective small nonisomorphic bimeromorphic maps with connected fibers and whose models remain normal irreducible globally Weil rational-factorial compact Kähler fourfolds with rational Cartier adjoints. The coefficients and discrepancy bound permit generalized log canonical, not only generalized canonical, pairs.

Outline of the argument

Manuscript 1 (family 056, 2026-10-05): Relative programs and special termination reduce the sequence to controlled terminal pieces while preserving the nef datum and actual adjoints. Analytic cycle ranks remove divisorial losses, and strict mixed-Hodge weight comparisons bound the residual surface changes, leaving only finitely many possible steps.

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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