Determine whether, for every countable family of hyperplanes H_j_j∈N in general position in P^m and every linearly nondegenerate holomorphic curve f colonC→P^m of finite lower order, the Nevanlinna defects satisfy ∑_j=1^∞delta_f(H_j)^1/3<∞.

Endpoint Nevanlinna defect problem

Determine whether, for every countable family of hyperplanes $\{H_j\}_{j\in\mathbb{N}}$ in general position in $\mathbb{P}^m$ and every linearly nondegenerate holomorphic curve $f\colon\mathbb{C}\to\mathbb{P}^m$ of finite lower order, the Nevanlinna defects satisfy $\sum_{j=1}^{\infty}\delta_f(H_j)^{1/3}<\infty$ .

Progress summary

Solved

A new unrefereed manuscript claims to settle the long-standing higher-dimensional question, but its proof has not yet been independently confirmed.

The problem concerns the endpoint exponent for countably many hyperplanes in general position, extending the scalar theorem of Weitsman to higher-dimensional holomorphic curves. The latest manuscript claims that the sharp exponent is $1/3$ .

August 2026 endpoint result

A manuscript titled “Sharp Summability of Nevanlinna Defects for Finite-Lower-Order Holomorphic Curves” claims to establish the endpoint exponent $1/3$ for countably many hyperplanes in general position, thereby settling the higher-dimensional question. The manuscript is explicitly unrefereed, so this is a claimed resolution rather than a verified theorem.

Current status (as of August 2026): The endpoint $1/3$ result is claimed in an unrefereed manuscript, while independent verification and the status of the proof remain open.

Sources

arxiv.org

Sources & referencesView supporting material

Primary source

arXiv

Additional references

Sharp Summability of Nevanlinna Defects for Finite-Lower-Order Holomorphic Curves — arXiv

complex-analysis nevanlinna-theory

5 hours ago