Pith. sign in

REVIEW 2 minor 18 references

Degenerate Second Main Theorems for Holomorphic Curves in Different Geometric Settings

T0 review · 0 major / 2 minor · reviewed 2026-05-24 · grok-4.3

Challenge this review Download PDF Read on arXiv ↗

Pith's one-line read Holomorphic curves into a projective subvariety satisfy second main theorems with truncation levels independent of the number of hypersurfaces.

desk verdict The paper claims second main theorems with truncation levels independent of hypersurface count and an improved defect bound, across four geometric settings plus a Schmidt subspace theorem application. read the letter →

arxiv 2406.02371 v2 submitted 2024-06-04 math.CV

Si Duc Quang , Nguyen Van An , Tran An Hai This is my paper
classification math.CV
keywords holomorphiccurvessecondmaintheoremsubgeneralpositionprojectivesubvarietydefectrelationNevanlinnatheorySchmidtsubspace
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper establishes degenerate second main theorems for holomorphic curves mapping into a projective subvariety V of dimension k inside projective space. The curves intersect hypersurfaces placed in N-subgeneral position with respect to V, where N exceeds k. Explicit truncation levels for the associated counting functions are derived that remain fixed regardless of how many hypersurfaces appear. These theorems hold for entire curves on the complex plane, curves on annuli, curves on discs with finite growth index, and curves on Kähler manifolds. The work also produces an improved total defect bound and yields a corresponding form of Schmidt's subspace theorem for families of homogeneous polynomials in subgeneral position.

What carries the argument

The N-subgeneral position condition of the hypersurfaces with respect to the subvariety V, which carries the argument by enabling truncation levels independent of hypersurface count.

What would settle it

A holomorphic curve into V that intersects hypersurfaces in N-subgeneral position but requires truncation levels that grow with the number of hypersurfaces, or that violates the improved total defect bound.

Watch

Extended reading notes

Core claim

We establish second main theorems for holomorphic curves into a projective subvariety V ⊂ P^n(C) of dimension k, intersecting hypersurfaces in N-subgeneral position with respect to V (N > k). Our results provide explicit truncation levels for the counting functions that are independent of the number of hypersurfaces. The theorems are obtained in several settings, including holomorphic curves on C, annuli, complex discs with finite growth index, and Kähler manifolds. We obtain a total defect bound that improves upon the previously known results. As an application, we establish a corresponding form of Schmidt's subspace theorem for families of homogeneous polynomials in subgeneral position.

Load-bearing premise

The hypersurfaces must satisfy the N-subgeneral position condition with respect to V where N exceeds the dimension k of V.

Editorial extensions

If this is right

  • Explicit truncation levels for counting functions stay fixed no matter how many hypersurfaces are present.
  • A total defect bound is obtained that improves on earlier results in the same geometric settings.
  • A form of Schmidt's subspace theorem holds for families of homogeneous polynomials in subgeneral position.
  • The theorems apply uniformly across entire curves on C, curves on annuli, curves on discs with finite growth index, and curves on Kähler manifolds.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The fixed truncation may simplify explicit computations of defects for curves in higher-dimensional targets.
  • The subspace theorem application could connect value distribution results to arithmetic statements about polynomial families.
  • The approach might extend to other value distribution problems where subgeneral position replaces general position.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 2 minor

Summary. The paper establishes second main theorems for holomorphic curves into a projective subvariety V ⊂ ℙⁿ(ℂ) of dimension k, intersecting hypersurfaces in N-subgeneral position with respect to V (N > k). The results provide explicit truncation levels for the counting functions that are independent of the number of hypersurfaces. Theorems are proved in several settings: entire curves on ℂ, annuli, complex discs with finite growth index, and Kähler manifolds. A total defect bound is obtained that improves on prior results, with an application to a form of Schmidt's subspace theorem for families of homogeneous polynomials in subgeneral position.

Significance. If the central derivations hold, the work would advance Nevanlinna theory by furnishing truncation levels independent of the number of hypersurfaces under the N-subgeneral position hypothesis (N > k), yielding improved defect bounds across multiple geometric settings. The explicit application to Schmidt's subspace theorem for homogeneous polynomials constitutes a concrete strength, as it supplies a falsifiable Diophantine statement directly tied to the new truncation results.

minor comments (2)
  1. Abstract, line 1: 'projective subvary' appears to be a typographical error for 'projective subvariety'.
  2. The manuscript would benefit from an explicit statement, perhaps in the introduction or a dedicated comparison section, of how the new total defect bound quantitatively improves on the best previously known constants under the same N-subgeneral position hypothesis.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive evaluation of the manuscript, including the recommendation for minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity in derivation chain

full rationale

The paper claims to establish second main theorems with explicit truncation levels independent of the number of hypersurfaces and an improved total defect bound, across standard geometric settings. No load-bearing steps reduce by construction to fitted parameters, self-definitions, or self-citation chains; the N-subgeneral position hypothesis and growth conditions are standard external inputs. The application to Schmidt's subspace theorem is presented as a consequence rather than a renaming or tautology. This is the typical honest non-finding for a theorem-proving paper in complex geometry.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

Based solely on the abstract, the paper relies on standard background from Nevanlinna theory and complex geometry without introducing new free parameters or invented entities.

assumptions (1)
  • domain assumption Standard properties of holomorphic curves, Nevanlinna counting functions, and the definition of N-subgeneral position
    The theorems extend classical results in value distribution theory.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Degenerate Second Main Theorems for Holomorphic Curves in Different Geometric Settings." pith.science (2026). https://pith.science/paper/2406.02371

@misc{pith2026240602371,
  author       = {Pith},
  title        = {Pith review of: Degenerate Second Main Theorems for Holomorphic Curves in Different Geometric Settings},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2406.02371}},
  note         = {Machine review of arXiv:2406.02371}
}
abstract

We establish second main theorems for holomorphic curves into a projective subvary $V \subset \mathbb{P}^n(\mathbb{C})$ of dimension $k$, intersecting hypersurfaces in $N$-subgeneral position with respect to $V$ $(N > k)$. Our results provide explicit truncation levels for the counting functions that are independent of the number of hypersurfaces. The theorems are obtained in several settings, including holomorphic curves on $\mathbb{C}$, annuli, complex discs with finite growth index, and K\"ahler manifolds. We obtain a total defect bound that improves upon the previously known results. As an application, we establish a corresponding form of Schmidt's subspace theorem for families of homogeneous polynomials in subgeneral position.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

  1. [1]

    Cartan, Sur les z´ eroes des combinaisons lin´ earies de p fonctions h olomorphes donn´ ees, Mathe- matica 7 (1933), 5–31

    H. Cartan, Sur les z´ eroes des combinaisons lin´ earies de p fonctions h olomorphes donn´ ees, Mathe- matica 7 (1933), 5–31

    work page 1933

  2. [2]

    T. B. Cao and Z. S. Deng, On the uniqueness of meromorphic functions that share three or two finite sets on annuli , Proc. Indian Acad. Sci. 122 No. 2 (2012), 203–220. 20 SI DUC QUANG AND TRAN AN HAI

    work page 2012

  3. [3]

    Evertse and R

    J. Evertse and R. Ferretti, Diophantine inequalities on projective varieties, Internat. Math. Res. Notices 25 (2002) 1295–1330

    work page 2002

  4. [4]

    Khrystiyanyn and A

    A.Y. Khrystiyanyn and A. A. Kondratyuk, On the Nevanlinna theory for meromorphic functions on annuli, I, Mat. Stud. 23 No. 01 (2005), 19–30

    work page 2005

  5. [5]

    A. Y. Khrystiyanyn and A. A. Kondratyuk, On the Nevanlinna theory for meromorphic functions on annuli. II, Mat. Stud. 24 No. 02 (2005), 57–68

    work page 2005

  6. [6]

    Nevanlinna, Einige Eideutigkeitss¨ atze in der Theorie der meromorphen Funktionen, Acta

    R. Nevanlinna, Einige Eideutigkeitss¨ atze in der Theorie der meromorphen Funktionen, Acta. Math., 48 (1926), 367–391

    work page 1926

  7. [7]

    E. I. Nochka, On the theory of meromorphic functions, Sov. Math. Dokl., 27 (1983), 377–381

    work page 1983

  8. [8]

    Noguchi, A note on entire pseudo-holomorphic curves and the proof of C artan-Nochka’s theorem, Kodai Math

    J. Noguchi, A note on entire pseudo-holomorphic curves and the proof of C artan-Nochka’s theorem, Kodai Math. J., 28 (2005), 336–346

    work page 2005

Show all 18 references
  1. [9]

    Noguchi, T

    J. Noguchi, T. Ochiai, Introduction to Geometric Function Theory in Several Compl ex Variables , Trans. Math. Monogr. 80, Amer. Math. Soc., Providence, Rhode I sland, 1990

    1990

  2. [10]

    H. T. Phuong, I. Padaphet, A uniqueness theorem for holomorphic curves on annulus shar ing hy- persurfaces, Complex Var. Elliptic Equ., (2023) 1–20. doi:10.1080/17476933.2023 .2234830

    2023 doi

  3. [11]

    S. D. Quang and D. P. An, The Second Main Theorem for holomorphic curves into a comple x projective space, Acta Math. Vietnamica, 38 (2017), 455–470

    2017

  4. [12]

    S. D. Quang, Generalizations of degeneracy second main theorem and Schm idt’s subspace theorem , Pacific J. Math., 318 (2022), No. 1, 153–188

    2022

  5. [13]

    S. D. Quang, An effective function field version of Schmidt’s subspace the orem for projective varieties, with arbitrary families of homogenous polynomials , J. Number Theory, 241 (2022) 563–580

    2022

  6. [14]

    S. D. Quang, Defect relation for holomorphic maps from complex discs int o projective varieties and hypersurfaces, arXiv:2404.18111v1 [math.CV]

  7. [15]

    Ru, A defect relation for holomorphic curves interecting hyper surfaces, Amer

    M. Ru, A defect relation for holomorphic curves interecting hyper surfaces, Amer. J. Math. 126 (2004), 215–226

    2004

  8. [16]

    Ru, Holomorphic curves into algebraic varieties , Ann

    M. Ru, Holomorphic curves into algebraic varieties , Ann. Math. 169 (2009), 255–267

    2009

  9. [17]

    Shiffman, On holomorphic curves and meromorphic maps in projective sp ace, Indiana Univ

    B. Shiffman, On holomorphic curves and meromorphic maps in projective sp ace, Indiana Univ. Math. J., 28 (1979), No. 4,627–641

    1979

  10. [18]

    L. Yang, Y. Zhu, Second main theorem for holomorphic curves on annuli with ar bitrary families of hypersurfaces, Electronic Research Archive, 32 (2024), 1365–1379. doi: 10.3934/era.2024063 Department of Mathematics, Hanoi National University of Ed ucation, 136-Xuan Thuy, Cau G...

    2024 doi

Pith tools

Reviewed May 24, 2026 · model on record in the stance chip above.