The Campana-Peternell conjecture on Kodaira dimension

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.

This result uses the release's log abundance theorem as an input, and the release contradicts itself on whether that theorem is unconditional (24 September abstract) or a conditional resolution resting on logarithmic Iitaka subadditivity (5 October appendix). The entry is Contested with it. 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.