Result 038, Algebraic and complex geometry

Fujita’s freeness conjecture

Proves Fujita's freeness conjecture at its sharp bound in every dimension: for a smooth projective complex variety X of dimension n and an ample line bundle L, the adjoint KX+mLK_X+mL is globally generated for every integer m≥n+1m\ge n+1.

Proof

The bigger picture

Why it matters

Geometric constructions can fail at points where all the available coordinates vanish. This manuscript claims a sharp guarantee that a standard family of coordinate systems avoids that failure on smooth complex projective varieties, in every dimension.

What changes?

The reported statement applies to every smooth complex projective variety of dimension n and every ample line bundle L. Combining the canonical line bundle with m copies of L gives global generation for every integer m at least n+1. A projective variety is a space defined by homogeneous polynomial equations. The canonical bundle records top-degree differential forms; ample means sufficiently high powers yield projective embeddings. Global generation means there is a global section nonzero at each point.

What does that help mathematicians do?

The claimed guarantee would let researchers use global sections of these adjoint bundles as projective coordinates without leaving any points undefined. The threshold depends only on dimension, not on the particular variety or ample bundle. Since the reported bound is sharp, it also identifies how far such a uniform guarantee can go. It guarantees an everywhere-defined map, not necessarily an embedding that separates all points and directions.

Are there practical applications?

Its immediate value is foundational. In algebraic and complex geometry, the claimed bound would provide a reusable criterion for constructing maps from adjoint line bundles, replacing case-by-case checks for points where all sections vanish. This is an existence statement, not an algorithm for computing those sections or maps.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Fujita's freeness conjecture

September 23, 2026 25 pages

We prove Fujita's freeness conjecture. If X is a smooth complex projective variety of dimension n and L is an ample line bundle, then KX+mLK_X+mL is globally generated for every integer m≥n+1m\geq n+1.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Fujitas-freeness-conjecture-September-23-2026,
  author = {{OpenAI}},
  title = {{Fujita's freeness conjecture}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Fujitas-freeness-conjecture-September-23-2026/Fujitas-freeness-conjecture-September-23-2026.pdf}{OAI:Fujitas-freeness-conjecture-September-23-2026}},
  year = {2026}
}

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.