Let X be a smooth complex projective n-dimensional variety, let F be an ample vector bundle on X that is a subsheaf of wedge^pT_X, and let p>0. Assume that F=wedge^pE for some ample vector bundle E. Kovács's exterior-power characterization conjecture. Then either X simeq P^n, or p=n and X simeq Q_n. This generalizes the Araujo–Druel–Kovács characterization theorem to higher-rank ample exterior-power subsheaves. The supplied context attributes the conjecture to S. Kovács and says that it was mentioned by K. Ross; no resolution is given.

Kovács's exterior-power characterization conjecture

About 16 years old · traced to 2010

Let $X$ be a smooth complex projective $n$ -dimensional variety, let $\mathcal{F}$ be an ample vector bundle on $X$ that is a subsheaf of $\wedge^pT_X$ , and let $p>0$ . Assume that $\mathcal{F}=\wedge^p\mathcal{E}$ for some ample vector bundle $\mathcal{E}$ . Kovács's exterior-power characterization conjecture. Then either

X\simeq \mathbb{P}^n,

or $p=n$ and

X\simeq Q_n.

This generalizes the Araujo–Druel–Kovács characterization theorem to higher-rank ample exterior-power subsheaves. The supplied context attributes the conjecture to S. Kovács and says that it was mentioned by K. Ross; no resolution is given.

References

Primary source

Yuting Liu, “Positivity of exterior products of tangent bundles and their subsheaves”, arXiv:2310.19315 (2023).

Additional references

2 papers in this index state this conjecture (2010–2023). The statement above is taken from the most recent of them; the others are arXiv:1012.2043.

Posted 1 month ago · edited 3 weeks ago

Progress summary

Refreshed 3 weeks ago

Claimed solved

A January 2026 paper claims to prove the conjecture, but the proof has not been independently verified.

Attributed to S. Kovács and mentioned by K. Ross, the conjecture says that an ample exterior-power subsheaf of the tangent bundle forces $X$ to be projective space, except for the top-degree smooth-quadric case.

Known results

Ross (2010): proved the conjecture when $\rho(X)=1$ , and recorded the low-dimensional cases.
Liu: proved the case $n\geq 3$ , $p=2$ , and $\binom{r}{2}\geq n-1$ .
Positivity of Exterior Powers (2023): proved $X\simeq\mathbb{P}^{n}$ when $\mathcal{F}=\bigwedge^{2}\mathcal{E}$ has rank at least $n$ and $n\geq 3$ .
Araujo–Druel–Kovács (2007): established the related line-bundle characterization.

January 2026 claimed proof

The paper Kovács’ conjecture on characterization of projective space and hyperquadrics states that its Theorem 1.1 proves the full characterization: $X\simeq\mathbb{P}^{n}$ , or $p=n$ and $X\simeq Q_n$ . The claim is currently unverified.

Current status (as of September 2026): A January 2026 arXiv paper claims a complete proof; earlier partial cases are established, but the full conjecture remains unverified.

Sources

No solutions have been posted yet.