The Campana-Peternell conjecture on Kodaira dimension

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Revised edition (October 6, 2026): Conditional good minimal models for compact Kähler fourfolds evidence for this event

Input manuscript: Log abundance in characteristic zero evidence for this event

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VibeMathed record: The Campana-Peternell conjecture on Kodaira dimension evidence for this event

Schnell, Singular metrics and a conjecture by Campana and Peternell (arXiv 2202.01295) evidence for this event

The Campana-Peternell conjecture on Kodaira dimension evidence for this event

Corollary 1.2: if $m_0K_X-D$ is pseudo-effective for an effective Cartier (or, after clearing denominators, rational) divisor $D$ on a smooth connected projective complex variety $X$, then $\kappa(X)\ge\kappa(D)$. Corollary 1.3: if $f:X\to Y$ is a fibre space and $m_0K_X-f^*H$ is pseudo-effective with $H$ ample, then $\kappa(X)=\kappa(F)+\dim Y$ and $\ell rK_X-f^*H$ has sections for large $\ell$ (Schnell's general fibre-space conclusion). Both use canonical nonvanishing in every dimension from the family's abundance manuscript. The zero-Kodaira case of Schnell's question was already proved by Zou (2025) and is not new here. parent of this event

The Campana-Peternell conjecture on Kodaira dimension parent of this event

The 5 October appendix calling projective log abundance conditional evidence for this event