The log abundance conjecture in characteristic zero (rational boundary, every dimension), with canonical nonvanishing

Theorem 1.1: for a normal projective lc pair $(X,B)$ over an algebraically closed field of characteristic zero, with $B$ effective rational and $K_X+B$ $\mathbb Q$-Cartier and nef, some positive Cartier multiple of $K_X+B$ is globally generated. The induction also gives good minimal models for lc pairs with pseudo-effective real adjoint and, over $\mathbb C$, canonical nonvanishing for smooth projective varieties in every dimension; consequences include rational slc abundance (Fujino-Gongyo) and finite generation of lc rings. Not shown: semiampleness with real boundaries, positive characteristic, or any bound on the multiple. The family's compact Kahler log abundance theorem assumes logarithmic Iitaka subadditivity (claimed in family 033) and is not part of this entry.

The release contradicts itself on this claim. The 24 September manuscript's abstract states that the rational-boundary log abundance conjecture is proved in every dimension. An appendix of the release's own 5 October manuscript on compact Kahler fourfolds calls the same theorem "a positive conditional resolution of the log abundance conjecture. It is not an unconditional abundance theorem", resting on a logarithmic Iitaka subadditivity assumption that it says follows from a lemma elsewhere in the release. Until OpenAI clarifies which reading is meant, or a specialist checks the chain, the entry is Contested. Theorem 1.1: for a normal projective lc pair $(X,B)$ over an algebraically closed field of characteristic zero, with $B$ effective rational and $K_X+B$ $\mathbb Q$-Cartier and nef, some positive Cartier multiple of $K_X+B$ is globally generated. The induction also gives good minimal models for lc pairs with pseudo-effective real adjoint and, over $\mathbb C$, canonical nonvanishing for smooth projective varieties in every dimension; consequences include rational slc abundance (Fujino-Gongyo) and finite generation of lc rings. Not shown: semiampleness with real boundaries, positive characteristic, or any bound on the multiple. The family's compact Kahler log abundance theorem assumes logarithmic Iitaka subadditivity (claimed in family 033) and is not part of this entry.