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The generalized Mukai conjecture
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Abstract
We prove the generalized Mukai conjecture: every positive-dimensional smooth complex Fano manifold of dimension n, Picard number ρ, and pseudoindex ι satisfies $\rho(\iota-1)\le n$. Equality holds precisely for the product of ρ copies of $\mathbb P^{\iota-1}$.
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Follow-up works
- Heath Pearson, The Mukai conjecture via Cox rings for smooth canonical ambient embeddings (arXiv:2604.25023v2, 7 Oct 2026). Version 2 cites this preprint, noting that "an unreleased OpenAI model claimed a solution of the (generalised) Mukai conjecture using Gromov-Witten invariants", and calls that approach "genuinely distinct" from its own Cox-ring proof for ambiently smooth Fano varieties; Remark 1.10 relates its stronger bound to the OpenAI claim.
Possibly relevant literature (found by the agent, not checked)
- Giuliano Gagliardi and Johannes Hofscheier, The generalized Mukai conjecture for symmetric varieties (arXiv:1412.6084, Trans. Amer. Math. Soc. 369 (2017)). Proves the generalized Mukai conjecture for symmetric varieties, one of the earlier special cases surveyed in the paper's introduction; the bibliography cites only the later 2025 spherical paper by Gagliardi, Hofscheier and Pearson.
- Paolo Bravi and Guido Pezzini, The generalized Mukai conjecture for spherical varieties with a reductive general isotropy group (arXiv:2501.04405, 2025-01-08). Establishes a conjecture of Gagliardi and Hofscheier implying the generalized Mukai conjecture for this spherical class, relevant to the paper's review of known equivariant cases.
- Joshua Enwright, Stefano Filipazzi, Yoshinori Gongyo, Joaquín Moraga, Roberto Svaldi, Chengxi Wang and Kiwamu Watanabe, Characterization of products of projective spaces via nef complexity (arXiv:2512.13637, Dec 2025). Characterizes products of projective spaces via nef complexity and proves Mukai's conjecture for smooth Fano varieties all of whose extremal contractions are of fiber type, which bears on the equality case of Theorem 1.1.
- Kento Fujita, Around the Mukai conjecture for Fano manifolds (arXiv:1409.8433, 2014/2015). Studies a refinement of the index version of Mukai's conjecture and classifies the near-extremal Fano manifolds for Picard number at most 3 or dimension at most 5, related to the paper's corollary on the index version.
- Kiwamu Watanabe, Smooth Fano varieties with pseudoindex equal to half of their dimension (arXiv:2411.00379, 2024-11-01; revised 2025-10-21). Classifies smooth complex Fano varieties with pseudoindex equal to half their dimension that admit a birational extremal contraction, a comparison point for the manuscript's claims on pseudoindex and extremal geometry.
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 063
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:The-Generalized-Mukai-Conjecture-September-24-2026,
author = {{OpenAI}},
title = {{The generalized Mukai conjecture}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/The-Generalized-Mukai-Conjecture-September-24-2026/article.pdf}{OAI:The-Generalized-Mukai-Conjecture-September-24-2026}},
year = {2026}
}Family 063: The generalized Mukai conjecture
Proves the generalized Mukai conjecture: every positive-dimensional smooth complex projective Fano manifold of dimension n, Picard number ρ and pseudoindex ι satisfies $\rho(\iota-1)\le n$, with equality exactly for $(\mathbb P^{\iota-1})^\rho$. Here the pseudoindex is the least anticanonical degree of a rational curve.
This is the only paper in this family.