Fujita’s freeness conjecture

Fujita’s conjecture predicts the sharp threshold for global generation of adjoint line bundles. It was formulated by Takao Fujita in 1985 and remains open in general.

Known results

• Curves: proved by Riemann–Roch.
• Surfaces: proved by Reider.
• Threefolds: freeness for $m\ge 4$ proved by Ein–Lazarsfeld (1993).
• Fourfolds and fivefolds: proved by Kawamata, and by Ye–Zhu in dimension $5$ (2020); the general case remains open for dimensions $6$ and above.

Recent claimed progress

A September 2026 preprint claims an explicit linear global-generation bound with constant $1.77629\ldots$, implying the simpler bound $m=2n$, but it does not prove the conjectured sharp threshold and is unrefereed. A November 12 announcement called a 2024 preprint a solution of the full conjecture; that claim is unverified.

Current status (as of September 2026): the freeness conjecture is proved through dimension $5$; in dimensions $6$ and above it remains open, with the new linear bound and earlier claimed full solution unverified.