Viehweg's conjecture C(n,m)+ and Popa's logarithmic Iitaka-Viehweg inequality
Viehweg's conjecture strengthens Iitaka's subadditivity by the birational variation of the family: for a fibre space of smooth projective varieties with , . Kollar proved it for fibres of general type and Kawamata for fibres with good minimal models. Popa (Conjecture 3.8 of his note Conjectures on the Kodaira dimension) asked for the logarithmic version: for a projective fibre space of smooth quasi-projective varieties with and generic fibre , is ?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Birational geometry, Kodaira dimension and variation
- Posed by
- Mihnea Popa, Conjectures on the Kodaira dimension, Conjecture 3.8 (author version 2023; LMS LNS 489, 2025), generalising E. Viehweg's C(n,m)+ conjecture (1983)
- Year posed
- 2023
- Years open
- 3y
- Solved
- 2026-09-26
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 45 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: for a projective surjective morphism with connected fibres between smooth connected complex quasi-projective varieties with and geometric generic fibre , , with no good-minimal-model or abundance hypothesis on . Corollary 1.2 gives Viehweg's ordinary for projective complex fibre spaces with . Not covered: bases with , positive characteristic, and the additivity equalities for smooth families (a separate entry).
What the AI did
Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems. The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this family is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscripts are authored as OpenAI with no human author named. The README also cautions that unformalized results could have issues. This manuscript uses the reduced-SNC subadditivity theorem and full-period adjoint comparison of the family's orbifold subadditivity paper as stated inputs, so it stands or falls with that entry as well.
Verification
No independent mathematician has checked this yet. Checked here: Theorem 1.1 and Corollary 1.2 of 'Logarithmic Kodaira dimension and whole-fiber variation', read against Popa's Conjecture 3.8 in the 13 June 2023 author version (fetched and read here) and Viehweg's C(n,m)+ as Popa quotes it. The statement matches the conjecture, with variation defined through the field of definition of the whole geometric generic fibre. The proof (two constancy arguments, Hodge-line flatness, Demailly-Hacon-Paun extension, Hanamura descent) was not refereed. No Lean formalization. It depends on the orbifold subadditivity manuscript of the same family, which is also unreviewed.