Fujita’s freeness conjecture

On every smooth shape, n + 1 copies of a positive bundle are always enough for Fujita's freeness.

1 paper September 23, 2026 Explained by AI

Fujita's freeness conjecture, from 1987, concerns smooth shapes defined by polynomial equations. It predicts that K + mL has a section nonzero at each point once m reaches n + 1. Here n is the shape's dimension and L is any positive bundle. Before this paper it was known only up to dimension 5, with weaker bounds in general. This paper claims to prove the conjecture in every dimension, at the sharp value n + 1.

First step

1987

Takao Fujita, the conjecture

Significance

Major

If the proof holds, it settles a central conjecture of algebraic geometry at its sharp bound.

Trust

Claimed, not checked

Field

Algebraic and complex geometry

1 paper

Smallest m proven to work in dimension 10

Year	Who	Value
1995	Angehrn and Siu	56
2026	Han (preprint)	18
2026	This paper	11

Lower is better. 11 is the lowest possible value, as projective space shows. Values computed from each paper's stated bound with n = 10.

The explanation

Picture the projective plane. Its points are triples [x : y : z], not all zero. Scaling all three gives the same point. So [1 : 2 : 3] and [2 : 4 : 6] are one point. A polynomial whose terms all have the same degree has a clear zero set there. Now ask a simple question. Can you pick polynomials of a given degree so that, at every point, at least one of them is nonzero?

For degree 0 the answer is yes. The constant 1 never vanishes. For degree 1 there are three choices: x, y and z. At every point at least one coordinate is nonzero, so again yes. Degree 2 offers 6 polynomials, and degree 3 offers 10. For degree −1 there are no polynomials at all, so the answer is no.

Fujita's freeness conjecture asks this question on every smooth shape. A smooth projective variety [a shape cut out by polynomial equations, with no corners or pinches] has a dimension n. The plane has n = 2. A line bundle is a rule saying which function-like objects, called sections, live on the shape. It is ample when it is positive in a standard sense. Each shape also carries one built-in line bundle, the canonical bundle K. Adding m copies of an ample bundle L to K gives the adjoint bundle K + mL. It is globally generated [free] when, at every point, some section is nonzero there.

In the plane, take L to be the bundle of degree-1 polynomials. Then the sections of K + mL are the polynomials of degree m − 3. So m = 3 works and m = 2 gives nothing. In 1987 Takao Fujita predicted that m = n + 1 always works. The plane shows that n + 1 cannot be lowered. The paper states the same sharpness for projective space of every dimensionSource 1. The bundle L itself need not have any nonzero sections.

The known cases climbed slowly, one dimension at a time. Curves follow from the Riemann–Roch formula. Surfaces follow from Reider's theorem of 1988. Ein and Lazarsfeld did dimension 3 in 1993. Kawamata did dimension 4 in 1997, and Ye and Zhu dimension 5 in 2020. In every dimension, Angehrn and Siu proved m ≥ (n² + n + 2)/2 is enough in 1995. In dimension 10 that means 56, against the predicted 11. Later work brought it to order n^(4/3), then n(log log n + 2.34). A 2026 preprint by Jingjun Han reached a linear bound, about 1.776 times n. In dimension 10 that gives 18. In 2024 Tsz On Mario Chan announced a proof of a stronger form. That preprint was withdrawn after a step failed.

This paper claims to prove the full conjectureSource 2. Take any smooth complex projective shape of dimension n and any ample L. Then K + (n + 1)L is globally generated. It also claims the same for every larger mSource 3.

The standard method works one point at a timeSource 4. To get a nonzero section at a point, it builds a carefully placed singularity there. A vanishing theorem then lifts a value at the point to a section on the whole shapeSource 5. The trouble is where the singularity concentrates. It often settles along a curve or surface through the point, called a center, not on the point alone. That center can itself be badly shaped, and cutting it down costs positivitySource 6. The paper chooses its construction by minimizing a single average. It then claims an estimate that holds equally for all such centers of a given dimensionSource 7. It holds no matter how singular or how complicated the center is. This lets it push the center down to the point itself.

The result has limits worth keeping in view. It only promises a nonzero section at each point. It does not claim that sections tell points or directions apartSource 8. It does not touch the second, stronger part of Fujita's conjecture, about embedding the shapeSource 8. It is stated over the complex numbers.

How sure is this? No Lean formalization is listed for this paper, so no computer proof checker has examined it. No specialist review has been published. An earlier claimed proof of a stronger form was withdrawn, so a careful outside check matters here.

What it doesn't show. This does not prove that sections separate points. Nor does it prove the stronger part of Fujita's conjecture, about embedding the shape in projective space.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

Major if the proof holds. No Lean formalization is listed and no specialist review is published. A 2024 claimed proof of a stronger form was withdrawn, so caution is warranted.

Words used

Smooth projective variety: A shape cut out by polynomial equations, with no corners or pinches.
Line bundle: A rule saying which function-like objects, called sections, live on a shape.
Ample: Positive in a standard sense; some multiple of the bundle can place the shape inside projective space.
Canonical bundle: A line bundle that every smooth shape carries, written K.
Adjoint bundle: The canonical bundle plus m copies of an ample bundle, written K + mL.
Globally generated: At every point of the shape, some section is nonzero there. Also called free.
Center: The curve, surface or point where a constructed singularity concentrates.

The original paper

Fujita's freeness conjecture (PDF, opens in a new tab)
September 23, 2026

All quoted lines (8)

The exponent is sharp: for $X=\mathbb P^n$ and $L=\cO_{\mathbb P^n}(1)$ , the bundle $K_X+(n+1)L$ is trivial, whereas $K_X+nL$ has no nonzero sections.
Fujita's freeness conjecture Back to text
Let $X$ be a smooth connected projective variety over $\C$ of dimension $n\geq 1$ , and let $L$ be an ample line bundle on $X$ .
Fujita's freeness conjecture Back to text
If $X$ is a smooth complex projective variety of dimension $n$ and $L$ is an ample line bundle, then $K_X+mL$ is globally generated for every integer $m\geq n+1$ .
Fujita's freeness conjecture Back to text
The common vanishing strategy constructs controlled singularities at the prescribed point, reduces a suitable center to a point, and then lifts a nonzero fiber value.
Fujita's freeness conjecture Back to text
Kawamata–Viehweg vanishing then lifts a nonzero fiber value at $x$ .
Fujita's freeness conjecture Back to text
A recurring difficulty is that a positive-dimensional center can itself be singular, and cutting it down costs positivity.
Fujita's freeness conjecture Back to text
The estimate is uniform over all integral subvarieties of a fixed dimension, including singular subvarieties of unbounded degree.
Fujita's freeness conjecture Back to text
Global generation asks for a nonvanishing section at each point; it does not assert separation of points or tangent directions, or the very-ampleness part of Fujita's conjecture.
Fujita's freeness conjecture Back to text