Fujita’s freeness conjecture

1 · The essentials

The manuscript addresses a question about when geometric data provide enough usable choices everywhere on a space. The spaces considered are smooth, complex, and projective: they have no singular points and can be described by homogeneous polynomial equations. The question combines data intrinsic to the space with repeated copies of another ingredient satisfying a positivity condition. The desired outcome is that, at every point, some globally defined choice is nonzero there. The manuscript claims this outcome whenever the number of copies is at least the dimension of the space plus one. The repository summary describes this threshold as sharp. The main result is not formalized in Lean and has not been formally verified.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine a collection of lamps whose combined light must reach every spot in a room. The desired conclusion is that no spot stays dark, though different lamps may illuminate different spots; this image does not capture the geometric hypotheses or explain the numerical threshold.

Why it matters

The claimed result answers Fujita’s freeness conjecture for every dimension under the stated hypotheses. It gives a single dimension-dependent threshold for global generation, rather than a condition stated separately for each space. The supplied abstract states the conclusion but does not provide the argument.

2 · Going further

The precise claim is this: let $X$ be a smooth complex projective variety of dimension $n$ , and let $L$ be an ample line bundle on $X$ . Then the adjoint line bundle $K_{X} + m L$ is globally generated for every integer $m \geq n + 1$ . Here $K_{X}$ denotes the canonical line bundle, and the additive notation means $K_{X} \otimes L^{\otimes m}$ . Global generation means that global sections span the fiber at every point; for a line bundle, this means that at each point there is a global section that does not vanish there.

The manuscript presents this statement as Fujita’s freeness conjecture, and the repository summary describes the bound as sharp in every dimension. Neither an example establishing sharpness nor an account of earlier partial results is supplied. The abstract gives no main idea, intermediate steps, or techniques of the argument, so those cannot be explained from this source. Only the main manuscript is provided; no companion manuscripts or relationships between them are described.

Useful notions

Smooth complex projective varieties
Line bundles and global sections
Ample line bundles
The canonical line bundle

What was known before / what is new

Before

The source identifies the question as Fujita’s freeness conjecture. It supplies no historical account, earlier bounds, or previously established special cases.

New

The manuscript claims global generation of $K_{X} + m L$ for every integer $m \geq n + 1$ , in every dimension, when $X$ is smooth, complex, and projective and $L$ is ample. The repository summary characterizes this as the sharp bound.

Limits and assumptions

The claim requires a smooth projective complex variety and an ample line bundle; the supplied text makes no claim outside these hypotheses. Its conclusion concerns global generation for integers $m \geq n + 1$ , not smaller values. No argument or sharpness example is included in the supplied abstract, and no further open questions are specified. The main result is not formalized in Lean and has not been formally verified.

Mini glossary

Smooth complex projective variety: A space described by homogeneous polynomial equations over the complex numbers that has no singular points.
Line bundle: A geometric structure assigning a one-dimensional vector space to each point, with these spaces fitting together locally.
Global section: A choice of an element of each fiber of a bundle, defined compatibly across the whole space.
Global generation: The property that global sections span the bundle’s fiber at every point.
Ample line bundle: A line bundle satisfying a positivity condition under which some positive tensor power gives a projective embedding.
Canonical line bundle: The line bundle of top-degree differential forms on a smooth variety, denoted here by $K_{X}$ .

Keywords : Fujita’s freeness conjecture · Adjoint line bundles · Global generation

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

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 $K_{X} + m L$ is globally generated for every integer $m \geq n + 1$ .

To be verified: the proof is written in the manuscript, but no software has checked it and no validation by independent mathematicians is established. It may contain errors.

Manuscripts in this family

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Fujita's freeness conjecture

23/09/2026

To be verified

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 $K_{X} + m L$ is globally generated for every integer $m \geq n + 1$ .

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