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Math-2435 · mathematics · Algebraic geometry

Kobayashi Pseudometric Vanishing on Calabi–Yau Manifolds

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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.
Does the Kobayashi pseudometric vanish identically on every compact Calabi–Yau manifold? In particular, must every such manifold be Kobayashi non-hyperbolic and contain an entire curve?

Formulation references: [S1], [S2].

Short English statement

Does the Kobayashi pseudometric vanish identically on every compact Calabi–Yau manifold? In particular, must every such manifold be Kobayashi non-hyperbolic and contain an entire curve?

Sources

Catalogue entry

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

Contributions 1

Math-2435-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2435

Manuscript 1 (family 042, 2026-09-23): Convex approximation makes every complex K3 surface Oka; the dense-entire-curve corollary in particular supplies entire curves through any point and tangent, giving zero Kobayashi pseudometric in that Calabi-Yau surface class. Higher-dimensional compact Calabi-Yau manifolds are not covered.

Kobayashi Pseudometric Vanishing on Calabi–Yau Manifolds

OpenAI's claim

Every complex K3 surface is claimed to satisfy the Oka property, including convex approximation in arbitrary source dimension. 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 any complex K3 surface $X$, any smooth Hermitian distance $d_X$, any $m\ge1$, any nonempty compact convex $K\subset\mathbb C^m$, neighborhood $U$ of $K$ and holomorphic $f:U\to X$, OpenAI claims \[ \forall\varepsilon>0\ \exists F:\mathbb C^m\to X\text{ holomorphic}:\quad\sup_{z\in K}d_X(F(z),f(z))<\varepsilon. \] This convex approximation property characterizes the Oka condition. It includes nonprojective K3 surfaces. A corollary claims, for every $x\in X$ and $0\ne v\in T_xX$, a holomorphic immersion $g:\mathbb C\to X$ with $g(0)=x$, $g'(0)=v$ and ordinary-topology-dense image; for projective $X$ the image is also Zariski dense.

Outline of the argument

Manuscript 1 (family 042, 2026-09-23): Local sprays and quantitative sewing propagate complete directions across long annular chains without losing transverse width. Polynomial changes of source coordinates convert parameter-dependent strip approximation into convex approximation, and the corrected period-orbit alternatives transfer the construction to every K3 period.

Formal verification

Family 042 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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