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Math-2404 · mathematics · Algebraic geometry

Rational Curves on Compact Kähler Manifolds

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Let $X$ be a compact Kähler manifold. Is $X$ covered by rational curves if and only if $K_X$ is not pseudoeffective? If $K_X$ is pseudoeffective but not nef, must $X$ contain a rational curve?

Formulation references: [S1], [S2].

Short English statement

Let $X$ be a compact Kähler manifold. Is $X$ covered by rational curves if and only if $K_X$ is not pseudoeffective? If $K_X$ is pseudoeffective but not nef, must $X$ contain a rational curve?

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2404-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2404

Manuscript 1 (family 051, 2026-09-23): For the hyperbolic compact Kähler subclass the theorem gives ample K and excludes rational curves. It does not prove the general equivalence or existence of a rational curve when a pseudo-effective K is non-nef.

Rational Curves on Compact Kähler Manifolds

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

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