Family 050 — Ample bundles without Griffiths-positive metrics on the quadric

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Family 050 · Algebraic and complex geometry
Importance 4/5 · Provisional editorial rating
Sources consulted 2026-10-12

1. The question

Algebraic geometry and differential geometry have different ways to say that a bundle is positive. Algebraic positivity asks whether enough sections eventually give a projective embedding. Differential positivity asks whether a metric has positive curvature in specified directions.

For line bundles on a smooth projective variety, these viewpoints agree. Griffiths’ conjecture asks whether the same is true for vector bundles: must an ample holomorphic vector bundle admit a smooth Hermitian metric with strictly Griffiths-positive curvature?

Family 050 contains one manuscript claiming counterexamples of rank two on the quadric surface $\mathbb P^1\times\mathbb P^1$. Every member of its explicit family is ample, but all sufficiently large members are asserted to admit no metric of the required kind.

The manuscript credits an earlier 2026 counterexample of Du–Xie on an abelian surface. Its contribution is a different construction on a rational surface, an explicit family with quantified algebraic positivity, and a limiting obstruction applying to every smooth strictly positive Hermitian metric on the indicated members.

2. Background and essential definitions

Bundles and metrics

A rank-two holomorphic vector bundle $E$ attaches a copy of $\mathbb C^2$ to every point of a complex manifold, with holomorphic changes of coordinates between local trivializations. A Hermitian metric is a smoothly varying positive-definite Hermitian inner product on those fibers.

In a local frame, its squared norm has the form

$$\|v\|_h^2=\sum_{\alpha,\beta=1}^{2} h_{\alpha\bar\beta}(x)v_\alpha\overline{v_\beta}.$$

The dependence on the fiber variable $v$ is quadratic. This restriction will matter: more general homogeneous norms, called complex Finsler metrics, need not arise from an inner product.

The Hermitian metric determines the Chern connection, whose curvature records how the fiber geometry varies around infinitesimal loops in the base. Strict Griffiths positivity requires a positive curvature value for every nonzero base tangent direction $\xi$ and every nonzero fiber vector $v$. In local coefficients, the condition is

$$\sum_{j,k,\alpha,\beta} c_{j\bar k\alpha\bar\beta} \xi_j\overline{\xi_k}v_\alpha\overline{v_\beta}>0.$$

Semipositivity replaces $>0$ by $\ge0$. Excluding strictly positive metrics does not automatically exclude semipositive ones.

Ampleness for a vector bundle

Let $\mathbb P(E)$ parametrize one-dimensional quotients of the fibers of $E$. It carries a tautological quotient line bundle $\mathcal O_{\mathbb P(E)}(1)$. The vector bundle $E$ is ample when this line bundle is ample, meaning some positive tensor power gives a projective embedding.

This definition tests the geometry of the projectivized bundle. It does not initially specify a quadratic norm on $E$ itself. A strictly Griffiths-positive Hermitian metric implies ampleness; the conjecture asks for the converse.

A Finsler metric can have the homogeneity of a norm without the quadratic formula above. Positivity results allowing such metrics, or metrics on a determinant twist or symmetric power of $E$, are therefore not automatically statements about the original Hermitian problem.

The base and its power maps

The quadric surface is

$$S=\mathbb P^1\times\mathbb P^1.$$

Its two projections give two families of projective lines, called rulings. The line bundle $\mathcal O_S(a,b)$ has degree $a$ along one ruling and $b$ along the other. Put $P=\mathcal O_S(1,1)$, which is ample.

The map $f_m:S\to S$ raises homogeneous coordinates in each factor to their $m$-th powers. It is finite of degree $m^2$, and

$$f_m^*P\simeq P^{\otimes m}.$$

Pullback changes the scale of the bundle’s geometry, while the family below adds only one fixed positive twist $P$.

3. Development history and previous partial results

Hartshorne’s theory of ample vector bundles organized algebraic positivity through symmetric powers, generation, and the tautological line bundle. Griffiths’ differential-geometric formulation asked for a bridge from such algebraic positivity to a curvature condition on a Hermitian metric.

The bridge is established for line bundles. It is also established for vector bundles on compact complex curves, through work of Umemura and Campana–Flenner, with later analytic approaches including Murakami’s work. Thus rank two and base dimension two are the first possible dimensions for the phenomenon claimed here.

Other positive results modify the bundle. Berndtsson proved Nakano positivity for $E\otimes\det E$ when $E$ is ample. Nakano positivity is stronger than Griffiths positivity, but the determinant twist is part of the conclusion. Liu–Sun–Yang obtained further positivity statements for symmetric powers and determinant-twisted bundles. These results do not simply remove the twist and produce the conjectured metric on $E$.

Kobayashi characterized ampleness through negatively curved strongly pseudoconvex Finsler structures on the dual bundle. The freedom to use a nonquadratic norm is crucial. Demailly’s later analytic systems and curvature functionals sought to recover the more restrictive Hermitian structure.

A recent obstruction of Lempert concerns a prescribed equivariant rule for constructing a Hermitian metric from a tautological metric. As M1 notes, those examples still admit positive Hermitian metrics by other means. Failure of one construction rule is weaker than nonexistence of every positive metric.

Most directly, Du–Xie’s preprint of 22 September 2026 states an ample rank-two bundle on an abelian surface without any smooth Griffiths-semipositive Hermitian metric. M1 explicitly credits that earlier claim and gives a different family on the quadric, with the weaker metric exclusion of strict positivity.

A subsequent Liu–Wan preprint of 29 September 2026 states examples on every Hirzebruch surface with additional stability and a stronger Finsler-curvature exclusion. Its curvature condition is on the original bundle, whereas Kobayashi’s ampleness characterization uses negative curvature on the dual. These are related developments with different conclusions and constructions.

4. What the new manuscripts claim

Theorem 1.1 of M1 asserts an explicitly constructed rank-two algebraic bundle $G$ on $S$ for which

$$E_m=f_m^*G\otimes P$$

is ample for every positive integer $m$. It further claims an integer $m_0$ such that for every $m\ge m_0$, the bundle $E_m$ has no smooth Hermitian metric with strictly Griffiths-positive curvature.

The quantifiers differ on the two sides. Ampleness holds for all $m\ge1$; metric nonexistence is asserted for all sufficiently large $m$. The theorem does not identify which of the finitely many smaller values admit such metrics.

The ampleness statement comes with a numerical bound. Proposition 3.1 states that

$$\mathcal O_{\mathbb P(E_m)}(n) \quad\text{is very ample whenever }n\text{ is even and }n-1\ge3m.$$

“Very ample” means the line bundle itself supplies a projective embedding. This is a concrete statement about enough algebraic sections. In contrast, the threshold $m_0$ is obtained by compactness and is ineffective: the manuscript gives no explicit numerical value.

The excluded metrics are smooth, Hermitian, and strictly Griffiths-positive. Singular metrics, semipositive metrics, nonquadratic Finsler structures, and metrics on modified bundles are different questions unless specifically addressed by another theorem.

5. Why the advance matters

Importance: 4/5. If correct, the construction would exhibit a broad family separating algebraic ampleness from smooth Hermitian curvature positivity on a particularly simple surface.

The central conjecture already has an earlier counterexample claim credited in the source. This family’s significance is therefore its specific rational base, explicit pullback-and-twist construction, all-parameter ampleness bound, and general limiting argument. It should not be described as having sole priority for the negative answer.

The geometric lesson is substantial: plenty of sections of the tautological bundle need not organize themselves into a smoothly varying quadratic norm with positive curvature on the original vector bundle. Fiberwise quadratic structure is additional information, not a harmless choice of notation.

The result would also distinguish flexibility gained by determinant twists or Finsler norms from the original metric problem. Those positive theories can remain valid while the Hermitian converse fails.

6. The main idea

Encode incompatible symmetries in a bundle

Choose two projective transformations $A,B$ of $\mathbb P^1$, each of order three, with no common fixed point, but such that their product acts, after a coordinate change, like

$$z\longmapsto\lambda z,\qquad |\lambda|>1.$$

The manuscript gives explicit matrix lifts and arranges finite orders for the relative lifts needed at graph intersections. Consider the three graphs of the identity, $A$, and $B$ inside $S$.

The bundle $G$ is built from pairs of sections of two line bundles, one positive along each ruling, required to agree along those graphs through prescribed identifications. This is an elementary transformation. Compatibility at intersections makes the construction locally free rather than merely a singular sheaf.

Two quotient-line-bundle sequences then control symmetric powers of $G$. Alternating their filtrations and adding the positive twist supplies the stated very-ampleness bound for every $E_m$.

Positive metrics would produce a nonzero quadratic limit

Assume positive metrics exist for infinitely many $m$. Their dual squared norms are plurisubharmonic: restriction to every holomorphic disc has the subharmonic mean-value property.

M1 averages these norms over the finite power maps and normalizes them. The twisting factor becomes asymptotically negligible. Its compactness argument produces a finite nonnegative plurisubharmonic function on $G^*$ that is a Hermitian quadratic form on every fiber and is not identically zero.

The limit may lose rank, so it is permitted to be only a seminorm. What must survive is nonzero mass and quadratic behavior even over the graph divisor. An almost-everywhere statement alone would not retain the relations imposed there.

The limit would force an impossible invariant measure

On either dual line subbundle, compactness of the opposite projective-line direction forces the limiting seminorm to be independent of that direction. The graph identifications therefore produce one nonzero seminorm on a negative line bundle over $\mathbb P^1$, invariant under both $A$ and $B$.

Taking curvature in the sense of distributions yields a finite positive measure of nonzero total mass. Invariance under the expanding product $AB$ forces that measure to be supported on its two fixed points: successive disjoint annuli otherwise all have the same positive mass, contradicting finiteness.

Both order-three transformations preserve this support. An order-three permutation of at most two points must fix each point, so they would have a common fixed point, contrary to their construction. This excludes the limiting quadratic object and hence excludes an infinite sequence of positive-metric parameters. Only finitely many small $m$ can remain.

7. Scope, limitations, and open questions

The theorem concerns the stated rank-two family on $\mathbb P^1\times\mathbb P^1$. It does not assert that every ample rank-two bundle on that surface lacks positive metrics; direct sums of suitable positive line bundles provide familiar positive examples.

There is no explicit value of $m_0$, so the claim does not identify a numerically specified first counterexample in the family. The explicit construction of $G$ and the effective very-ampleness bound do not make the metric threshold effective.

Strict positivity is essential to the reported exclusion. The theorem should not be upgraded to nonexistence of semipositive Hermitian metrics or every Finsler structure by borrowing conclusions from different examples in the cited literature.

Natural further questions concern the smallest failing parameter, additional hypotheses restoring Hermitian positivity, and the exact relationship among algebraic positivity, singular metrics, and nonquadratic curvature conditions.

8. Sources and further reading

M1’s Theorem 1.1 states the family and threshold; Proposition 3.1 gives the very-ampleness bound; Proposition 4.1 gives the limiting quadratic object; Section 5 supplies the invariant-measure obstruction.

The primary historical sources linked above separate the curve theorem, twisted-bundle results, Finsler characterization, and recent counterexample claims. Background on Hermitian vector bundles helps with the metric language. The distinction between a quadratic Hermitian norm and a more general homogeneous norm is the key point throughout.

Source manuscripts

• M1. OpenAI, Ample rank-two bundles on the quadric surface without Griffiths-positive metrics (September 24, 2026).

Sources consulted on 12 October 2026 at repository revision fd4aeeb. Download the manuscript citations.