The main result of an arXiv preprint is a summability theorem for a class of holomorphic curves: when the curve has finite lower order and satisfies the paper's linear nondegeneracy condition, the one-third powers of its Nevanlinna defects add up to a finite total across any countable family of hyperplanes in general position. The statement is about mathematical objects rather than an empirical sample. In practical terms, the result controls an entire countable list of defect values at once, provided the curve and the hyperplanes meet the theorem's requirements.
The result reaches an exact endpoint. The paper's examples show that the corresponding defect-power sums diverge for every exponent just below one-third, written as one-third minus epsilon for any positive epsilon. That makes the boundary important: the theorem gives convergence at one-third, while the examples indicate that the same conclusion cannot be extended to every smaller exponent under the paper's stated framework.
A bound that does not grow with the list
The countable-family conclusion is backed by a finite-family estimate. For every finite subfamily of hyperplanes in general position, the sum of the one-third-power defects is bounded by a finite constant associated with the curve and its growth condition. The paper says this constant is independent of how many hyperplanes are included and of their conditioning. That uniformity is what allows the finite estimates to support convergence for a countable family.
The claim is therefore stronger than a statement about one carefully chosen finite configuration. It says that every finite general-position subfamily fits under the same kind of bound, even as the family is enlarged, so long as the theorem's geometric assumptions remain in force. The paper does not turn that bound into a universal number: its value is tied to the curve and its lower-order condition.
Why one-third appears
The proof combines several layers of complex analysis and geometry. The authors describe a route that uses dimension reduction, regularization and allocation before applying Weitsman's estimates to a projective family. These steps are designed to transfer the endpoint question into a setting where the needed estimates can be assembled without depending on the size or conditioning of the finite hyperplane subfamily.
The authors attribute the particular exponent one-third to a cubic balance in the potential-theoretic part of the argument. That is the paper's explanation for why the proof lands at this power rather than merely proving convergence for a larger exponent. The sharpness examples then serve a different role: they test the boundary from below and show divergence whenever the exponent is reduced by any positive amount.
A claimed endpoint advance
The authors say the work resolves a long-standing problem in Nevanlinna theory. They also describe it as an extension of Weitsman's scalar endpoint theorem and as covering results attributed to Krutin for exponents strictly greater than one-third. Those are the authors' interpretations of the paper's place in the field; the supplied review does not independently reassess the earlier sources.
The same pattern for divisors
The paper extends the conclusion beyond hyperplanes. For an irreducible projective variety, it considers bounded-degree effective Cartier divisors in general position, with the curve contained in none of their supports. Under those conditions, the one-third-power defect series converges, and sums over finite divisor subfamilies have a uniform bound.
The divisor result is not presented as a separate numerical experiment or an unrelated estimate. Its proof reduces the problem to the hyperplane case by replacing the divisors with positive multiples of a common degree while preserving their supports, the noncontainment condition and the relevant defect. This reduction is the bridge between the projective hyperplane theorem and the bounded-degree extension.
A simpler special case for rational curves
The preprint also records a sharper finite-family statement for rational curves. When the hyperplanes are in general position, at most N of the hyperplane defects can be positive, and the sum of their one-third powers is bounded by N. This gives a direct bound in that special class rather than relying only on the broader finite-lower-order theorem.
Conditions that define the result
The theorem is conditional in several ways. Finite lower order is required; the supplied analysis says that analogous statements without a growth restriction fail. For hyperplanes, the curve must be linearly nondegenerate, the family must be in general position, and the curve must not be contained in any hyperplane. The divisor extension additionally requires an irreducible projective variety, positive dimension, bounded degrees, general position relative to the variety and noncontainment in each divisor support.
Those conditions matter when interpreting the headline result. The paper does not establish one-third-power summability for arbitrary-growth curves, for hyperplane families outside general position or for curves lying in a target hyperplane. Nor does it show that the finite constants are the same across all curves, varieties or degree bounds; their quantitative dependence is left unevaluated in the supplied analysis.
Because the work is a preprint, its contribution is a theoretical proof rather than empirical validation or a statistical estimate. The supplied record reports no numerical evaluation of the finite constants and no independent replication. The appropriate reading is therefore a conditional mathematical result: within the stated assumptions, the endpoint series is controlled, while broader growth conditions and sharper quantitative dependence remain open questions.
Funding and disclosure
The acknowledgments say that Song-Yan Xie reports support from Chinese national research programs, the National Natural Science Foundation of China and the Xiaomi Young Talents Program. The authors also disclose using artificial intelligence for drafting, language editing and consistency checks, while taking responsibility for the manuscript.
Paper data and sources
Original title: Sharp Summability of Nevanlinna Defects for Finite-Lower-Order Holomorphic Curves
Authors: Yun-Heng Du, Song-Yan Xie
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
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