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Math-368-A2 · revision r1 · on Math-368

OpenAI claim summary for Math-368

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 051, 2026-09-23): The main theorem assumes compact connected Kähler geometry and absence of nonconstant entire curves and concludes canonical ampleness, including projectivity; this is precisely the catalogue compact hyperbolic formulation.

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

Manuscript

Contents
  1. Kobayashi Conjecture: Hyperbolic Compact Kähler Manifolds Have Ample Canonical Bundle
  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

Kobayashi Conjecture: Hyperbolic Compact Kähler Manifolds Have Ample Canonical Bundle

OpenAI's claim

Every positive-dimensional compact Kähler manifold without nonconstant entire curves is claimed to be projective with ample canonical bundle. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
For a connected Kähler manifold $X$, assumed compact and having positive complex dimension, OpenAI claims \[ [\text{every holomorphic }\mathbb C\to X\text{ is constant}]\quad\Longrightarrow\quad K_X\text{ ample}. \] Consequently some positive tensor power of $K_X$ gives a holomorphic embedding into projective space. The hypothesis is absence of nonconstant entire curves, with no initial projectivity assumption.

Outline of the argument

Manuscript 1 (family 051, 2026-09-23): Hyperbolicity bounds derivatives of holomorphic discs, and a Hardy-space frame realizes center-fixed variations with an area-Hessian identity. Disc envelopes give nef canonical weights, while the complementary variation estimate forces a volume lower bound; these positivity controls lead to canonical ampleness.

Formal verification

Family 051 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