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

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

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Occurred: Sep 24, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The generalized Mukai conjecture on Picard number and pseudoindex of Fano manifolds · Generalized Mukai conjecture

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

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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.

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

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Bonavero-Casagrande-Debarre-Druel, Sur une conjecture de Mukai (2003)

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