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Math-425-A2 · revision r1 · on Math-425

OpenAI claim summary for Math-425

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

Manuscript 1 (family 041, 2026-09-23): The finiteness corollary requires second Betti number at least five; the fixed-dimension deformation-finiteness record imposes no Betti-number hypothesis.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Boundedness of Compact Hyperkahler Manifolds: Finiteness of Deformation Types in Each Dimension
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Mathematical claim 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Boundedness of Compact Hyperkahler Manifolds: Finiteness of Deformation Types in Each Dimension

OpenAI's claim

On a compact Kähler IHS manifold, a holomorphic line is claimed to be semiample whenever its Chern class is nonzero, nef, and isotropic, regardless of deformation type. 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 $X$ be an irreducible holomorphic symplectic manifold, required to be compact and Kähler, of complex dimension $2n$, $n\ge1$. Write $q_X$ for its Beauville--Bogomolov form and $\overline{\mathcal K_X}$ for the closed Kähler cone. For a holomorphic line bundle $L$ with nonzero $c_1(L)\in\overline{\mathcal K_X}$ and $q_X(c_1(L))=0$, OpenAI claims semiampleness and, more precisely, \[ L^m\simeq f^*A,\qquad f:X\twoheadrightarrow B,\qquad\dim_{\mathbb C}B=n, \] with $m>0$, connected fibers, normal projective $B$ and ample $A$. Every fiber component has dimension $n$ and the holomorphic symplectic form vanishes on its smooth locus. A separate corollary claims finitely many smooth deformation types in each dimension $2n$ among these manifolds with $b_2\ge5$.

Outline of the argument

Manuscript 1 (family 041, 2026-09-23): The prescribed lattice vector is retained through a degeneration that produces a special Lagrangian torus. Rotation makes that torus holomorphic on a nonprojective deformation, where the torus-fibration theorem supplies semiampleness; the fixed-class deformation result brings the conclusion back to the original manifold.

Formal verification

Family 041 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

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