Yuan’s Conjecture 5.4.1 on equidistribution

Let $K$ be a finitely generated field, let $X$ be a quasi-projective variety over $K$ , and let $\overline{L}$ be an adelically metrized line bundle satisfying the hypotheses of Yuan–Zhang's Conjecture $5.4.1$ . If $(x_n)_{n\geq 1}$ is a generic sequence of small points in $X(\overline{K})$ , then, for every non-trivial place $v$ of $K$ , the associated empirical probability measures $\frac{1}{N}\sum_{n=1}^{N}\delta_{x_n}$ on the analytification $X_v^{\mathrm{an}}$ converge weakly, as $N\to\infty$ , to the canonical measure $\mu_{\overline{L},v}$ determined by $\overline{L}$ : $\frac{1}{N}\sum_{n=1}^{N}\delta_{x_n}\mathrel{\Longrightarrow}\mu_{\overline{L},v}$ .

References

Primary source

arXiv

Additional references

Equidistribution for quasi-projective varieties over function fields — arXiv

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A recent preprint claims to prove the conjecture in the function-field setting, but the claim has not been independently verified.

Yuan and Zhang stated Conjecture $5.4.1$ in May 2021: generic sequences of small points on quasi-projective varieties over finitely generated fields should become equidistributed at each non-trivial place with respect to the canonical measure.

Known results

Yuan and Zhang, 2021: formulated the conjecture for quasi-projective varieties over finitely generated fields.
Yuan and Zhang, 2023: proved an equidistribution theorem for generic small points over number fields and one-variable function fields, including the arithmetically big case.

August 2026 preprint

The preprint Equidistribution for quasi-projective varieties over function fields claims to establish the conjectured function-field theorem and also proves an equidistribution result for arithmetically big line bundles. This is a claimed resolution, but independent verification is not recorded; its stated scope is limited to the preprint's hypotheses.

Current status (as of August 2026): The theorem is claimed proved in the stated function-field regime, while the full conjecture remains unverified and any cases outside the preprint's hypotheses remain open.

Sources

No solutions have been posted yet.