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The generalized Mukai conjecture on Picard number and pseudoindex of Fano manifolds

For a Fano manifold XX of dimension nn, Picard number ρX\rho_X and pseudoindex ιX=min⁡{−KX⋅C}\iota_X=\min\{-K_X\cdot C\} over rational curves CC, Mukai (1988) conjectured an index version of a bound on ρX\rho_X, and Bonavero, Casagrande, Debarre and Druel formulated the pseudoindex strengthening, proving it for n≤4n\le4 and for toric cases. Later work covered n=5n=5 (Andreatta-Chierici-Occhetta), ιX>n/3\iota_X>n/3 (Novelli-Occhetta, Novelli) and toric, horospherical and spherical varieties. Does every smooth complex Fano manifold of positive dimension satisfy ρX(ιX−1)≤n\rho_X(\iota_X-1)\le n, with equality only for (PιX−1)ρX(\mathbb P^{\iota_X-1})^{\rho_X}?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Algebraic geometry; Fano manifolds and rational curves
Posed by
Bonavero, Casagrande, Debarre and Druel, 'Sur une conjecture de Mukai' (Comment. Math. Helv. 2003), strengthening Mukai (1988)
Year posed
2003
Years open
23y
Solved
2026-09-24
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
32 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: every smooth connected complex projective Fano manifold of positive dimension nn has ρX(ιX−1)≤n\rho_X(\iota_X-1)\le n, with equality if and only if X≅(PιX−1)ρXX\cong(\mathbb P^{\iota_X-1})^{\rho_X}. Consequently Mukai's original index version also holds. The paper treats smooth complex Fano manifolds only: singular (Q\mathbb Q-factorial Gorenstein) Fano varieties beyond the known toric, horospherical and spherical cases, and positive characteristic, are not covered.

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with one fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author. The family has a single manuscript.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the conjecture as formulated by BCDD; it is the inequality and the equality classification in every dimension with no restriction on Picard number or geometry. The proof goes through small quantum cohomology, descendant lower bounds on moduli of stable maps and a recurrence argument, then the BCDD ordered-chain method at equality; it was not refereed here and leans on virtual-class technology. No Lean formalization exists for this family.

Sources

Changelog1 change

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