Kobayashi's conjecture: a compact Kahler manifold with no entire curves has ample canonical bundle

Theorem 1.1: if $X$ is a compact connected Kahler manifold of positive dimension and every holomorphic map $\mathbb C\to X$ is constant, then $K_X$ is ample and $X$ is projective. Corollaries: $K_X^{\otimes m}$ is globally generated for $m\ge n+2$ (using the release's Fujita freeness result), and a finite cover splits as a ball-quotient-type factor times a hyperbolic factor when the universal cover is semialgebraic. It does not cover singular spaces, non-Kahler compact complex manifolds, or the converse direction (Green-Griffiths-Lang type statements).

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Prior state unknown→proved

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Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Kobayashi's conjecture: a compact Kahler manifold with no entire curves has ample canonical bundle · Kobayashi canonical-ampleness conjecture

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VibeMathed record: Kobayashi's conjecture: a compact Kahler manifold with no entire curves has ample canonical bundle

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No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against Kobayashi's conjecture. It states that every compact connected Kahler manifold of positive dimension in which every holomorphic map from $\mathbb C$ is constant has ample $K_X$ , so a power of $K_X$ embeds $X$ projectively. That is the conjecture in the smooth compact Kahler category. The proof (extremal holomorphic discs with an area penalty, a boundary-normalised frame, a Hessian identity giving a volume lower bound, then Aubin-Yau and Morse inequalities) was not refereed. There is no Lean formalization. The singular and non-Kahler settings are outside its scope, as the paper says.

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Kobayashi's conjecture: a compact Kahler manifold with no entire curves has ample canonical bundle

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