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Math-2569 · mathematics · Algebraic geometry

The Oka property for every complex K3 surface

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For any complex K3 surface X with any smooth Hermitian distance, any $m\geq 1$, nonempty compact convex $K\subset C^m$, neighborhood U of K and holomorphic $f:U\to X$, can f be approximated on K to arbitrary uniform accuracy by a holomorphic map $C^m\to X$? Nonprojective K3 surfaces are included. This is the convex approximation form of the Oka property.

Short English statement

For any complex K3 surface X with any smooth Hermitian distance, any $m\geq 1$, nonempty compact convex $K\subset C^m$, neighborhood U of K and holomorphic $f:U\to X$, can f be approximated on K to arbitrary uniform accuracy by a holomorphic map $C^m\to X$? Nonprojective K3 surfaces are included. This is the convex approximation form of the Oka property.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2569-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2569

Manuscript 1 (family 042, 2026-09-23): 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.

The Oka property for every complex K3 surface

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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