Pith. sign in

REVIEW 1 cited by

Siu's analyticity theorem for positive pluriharmonic currents

T0 review · reviewed 2026-06-30 · grok-4.3

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read On projective manifolds, Lelong number superlevel sets of positive ddc-closed (1,1)-currents are analytic subsets of dimension at most 1.

desk verdict This extends Siu analyticity to positive ddc-closed (1,1)-currents on projective manifolds with a clean decomposition and effective-cone consequence. read the letter →

arxiv 2606.29680 v1 pith:WNG6FGK6 submitted 2026-06-29 math.CV math.AG

Tien-Cuong Dinh , Viet-Anh Nguyen This is my paper · ORCID
classification math.CVmath.AG
keywords positivecurrentsLelongnumbersanalyticsetsSiudecompositionprojectivemanifoldsddc-closedeffectiveconecohomologyclasses
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 authors prove an analyticity result for the Lelong numbers of positive ddc-closed currents of bidimension (1,1) on projective manifolds. For any c greater than zero, the set where the Lelong number is at least c is shown to be an analytic subset of dimension no more than 1. This property yields a decomposition of the current into a sum of multiples of analytic curves plus a remainder current with Lelong numbers vanishing except at countably many points. Consequently, the cohomology classes of currents that assign no mass to proper analytic sets lie in the dual of the effective cone in H^{1,1}.

What carries the argument

The Siu decomposition of the current into a sum of multiples of compact analytic curves plus a remainder current whose Lelong numbers vanish outside a countable set.

What would settle it

A counterexample consisting of a projective manifold and a positive ddc-closed bidimension (1,1) current whose set of points with Lelong number at least some c > 0 fails to be an analytic set of dimension at most 1.

Watch

Extended reading notes

Core claim

Let T be a positive ddc-closed current of bidimension (1,1) on a projective manifold X of dimension n. For every c > 0 the set of points of X where the Lelong number of T is larger or equal to c is an analytic subset of dimension at most 1 of X. Moreover, T equals the sum over i in I of lambda_i times the current of integration over V_i plus T_0, where the V_i form a finite or countable family of compact analytic curves, and T_0 is a positive ddc-closed current whose Lelong number vanishes outside a countable set. As a consequence, the cohomology class of every positive ddc-closed current of bidimension (1,1) on X which does not give mass to any proper analytic set belongs to the Poincaré du

Load-bearing premise

The manifold X is projective and the current T is positive and ddc-closed of bidimension (1,1).

Editorial extensions

If this is right

  • The superlevel sets of the Lelong number are analytic of dimension at most 1 for any c > 0.
  • The current admits a decomposition into a finite or countable sum of multiples of compact analytic curves plus a remainder current.
  • The cohomology class of any such current with no mass on proper analytic sets belongs to the Poincaré dual of the effective cone.

Reading between the lines

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

  • The result relies on the projectivity of the manifold to achieve the analyticity conclusion.
  • The restriction to bidimension (1,1) is necessary for the dimension bound of at most 1 on the analytic sets.
  • This extends Siu's original theorem from closed positive currents to the ddc-closed case in this setting.
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 / 0 minor

Summary. The manuscript proves that on a projective manifold X of dimension n, for any positive ddc-closed current T of bidimension (1,1), the superlevel sets {x ∈ X | ν(T,x) ≥ c} for c > 0 are analytic subsets of dimension at most 1. It establishes the Siu decomposition T = ∑_{i∈I} λ_i [V_i] + T_0, where the V_i are compact analytic curves (finite or countable family), λ_i ≥ 0, and the remainder T_0 has Lelong number vanishing outside a countable set. As a corollary, the cohomology class of any such current that charges no proper analytic set lies in the Poincaré dual of the effective cone of H^{1,1}(X,R).

Significance. If the proof holds, the result extends Siu's classical analyticity theorem from positive closed currents to the larger class of positive ddc-closed (pluriharmonic) currents of bidimension (1,1). This supplies a concrete decomposition and analyticity statement under the projectivity hypothesis, together with a direct link between the non-charging condition and membership in the dual effective cone. The bidimension-(1,1) restriction and projectivity assumption are explicitly required and match the scope of the claim.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the report and for accurately summarizing the main results of the manuscript. The significance statement correctly identifies the extension of Siu's theorem to positive ddc-closed currents of bidimension (1,1) under the projectivity assumption, along with the Siu decomposition and the consequence for cohomology classes. No specific major comments were listed in the report.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation self-contained

full rationale

The paper states and proves an analyticity theorem for Lelong sets of positive ddc-closed bidimension-(1,1) currents on projective manifolds, together with an associated Siu-type decomposition. The abstract and claim explicitly list the projectivity and bidimension hypotheses as necessary for the conclusion; no step reduces by definition to its own inputs, no fitted parameter is relabeled as a prediction, and no load-bearing uniqueness or ansatz is imported solely via self-citation. The central result is presented as a theorem whose proof constitutes the paper's content, with no internal reduction to prior fitted values or self-referential constructions visible at the level of the stated claims.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The result relies on the established theory of positive closed currents, Lelong numbers, and properties of projective manifolds in complex geometry. No new free parameters or invented entities are introduced in the abstract.

assumptions (2)
  • standard math Standard properties of positive ddc-closed currents and Lelong numbers on complex manifolds
    The theorem invokes the classical theory of currents developed by Lelong, Siu and others.
  • domain assumption Projective manifold X admits the necessary tools from algebraic geometry for the effective cone statement
    The consequence about the Poincaré dual of the effective cone requires the projective setting.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Siu's analyticity theorem for positive pluriharmonic currents." pith.science (2026). https://pith.science/paper/WNG6FGK6

@misc{pith2026260629680,
  author       = {Pith},
  title        = {Pith review of: Siu's analyticity theorem for positive pluriharmonic currents},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WNG6FGK6}},
  note         = {Machine review of arXiv:2606.29680}
}
abstract

Let $T$ be a positive $\ddc$-closed current of bidimension $(1,1)$ on a projective manifold $X$ of dimension $n.$ We show that for every $c > 0$ the set of points of $X$ where the Lelong number of $T$ is larger or equal to $c$ is an analytic subset of dimension at most $1$ of $X.$ Moreover, the following Siu decomposition holds $$T=\sum_{i\in I} \lambda_i[V_i] +T_0,$$ where $\{V_i\}_{i\in I}$ is a (possibly empty) finite or countable family of compact analytic curves in $X,$ $\lambda_i\in\mathbb{R}^+,$ and $T_0$ is a positive $\ddc$-closed current of bidimension $(1,1)$ on $X$ whose Lelong number vanishes outside a finite or countable set. As a consequence, the cohomology class of every positive $\ddc$-closed current of bidimension $(1, 1)$ on $X,$ which does not give mass to any proper analytic set, belongs to the Poincar\'e dual of the effective cone of $H^{1,1}(X,\mathbb{R}).$

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Quantitative hyperbolicity for complex manifolds via numerical invariants

    math.CV 2026-07 conditional novelty 8.0 of 10

    New numerical invariants called hyperbolic indices, defined via liftable directed currents on Demailly-Semple towers, give a sufficient condition for Kobayashi hyperbolicity and grow at least linearly in degree for ge...

Reference graph

Works this paper leans on

37 extracted references · 2 canonical work pages · cited by 1 Pith paper

  1. [1]

    6, 577-602

    Alessandrini, Lucia; Bassanelli, Giovanni: Plurisubharmonic currents and their extension across ana- lytic subsets.Forum Math.5(1993), no. 6, 577-602

    1993

  2. [2]

    Results Math.30(1996), no

    Alessandrini, Lucia; Bassanelli, Giovanni: Lelong numbers of positive plurisubharmonic currents. Results Math.30(1996), no. 3-4, 191–224

    1996

  3. [3]

    5, 567–595

    Bassanelli, Giovanni: A cut-off theorem for plurisubharmonic currents.Forum Math.6(1994), no. 5, 567–595

    1994

  4. [4]

    Math.147(2002), no

    Berndtsson, Bo; Sibony , Nessim: The B-equation on a positive current.Invent. Math.147(2002), no. 2, 371–428

    2002

  5. [5]

    Math.128(1997), no

    Bierstone, Edward; Milman, Pierre D.: Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariantInvent. Math.128(1997), no. 2, 207–302

    1997

  6. [6]

    (French)Ann

    Blanchard, Andr ´e: Sur les vari ´et´es analytiques complexes. (French)Ann. Sci. Ecole Norm. Sup. (3) 73(1956), 157–202

    1956

  7. [7]

    Boucksom, S ´ebastien: Divisorial Zariski decompositions on compact complex manifolds.Ann. Sci. Ecole Norm. Sup. (4)37(2004), no. 1, 45–76

    2004

  8. [8]

    Algebraic Geom

    Boucksom, S ´ebastien; Demailly , Jean-Pierre; Paun, Mihai; Peternell, Thomas: The pseudo-effective cone of a compact K ¨ahler manifold and varieties of negative Kodaira dimension.J. Algebraic Geom. 22(2013), no. 2, 201–248

    2013

Show all 37 references
  1. [9]

    Chiose, Ionu; Toma, Matei: Positive currents on non-k ¨ahlerian surfaces.Math. Res. Lett.30(2023), no. 2, 375–412

    2023

  2. [10]

    Manuscript available at http://www-fourier.ujf-grenoble.fr/„demailly/books.html,(2012)

    Demailly , Jean-Pierre: Complex Analytic and Differential Geometry . Manuscript available at http://www-fourier.ujf-grenoble.fr/„demailly/books.html,(2012)

    2012

  3. [11]

    of Math.(2)159(2004), no

    Demailly , Jean-Pierre; Paun, Mihai: Numerical characterization of the K ¨ahler cone of a compact K¨ahler manifold.Ann. of Math.(2)159(2004), no. 3, 1247–1274

    2004

  4. [12]

    Dinh, Tien-Cuong; Lawrence, Mark G.: Polynomial hulls and positive currents.Ann. Fac. Sci. Toulouse Math. (6)12(2003), no. 3, 317–334

    2003

  5. [13]

    Ann.354, (2012), no

    Dinh, Tien-Cuong; Nguy ˆen, Viˆet-Anh; Sibony , Nessim: Heat equation and ergodic theorems for Rie- mann surface laminations.Math. Ann.354, (2012), no. 1, 331-376

    2012

  6. [14]

    J.171(2022), no

    Dinh, Tien-Cuong; Nguy ˆen, Viˆet-Anh; Sibony , Nessim: Unique ergodicity for foliations on compact K¨ahler surfaces.Duke Math. J.171(2022), no. 13, 2627-2698

    2022

  7. [15]

    Sci.´Ecole Norm

    Dinh, Tien-Cuong; Sibony , Nessim: Regularization of currents and entropy .Ann. Sci.´Ecole Norm. Sup. (4)37(2004), no. 6, 959–971

    2004

  8. [16]

    Dinh, Tien-Cuong; Sibony , Nessim: Unique ergodicity for foliations inP 2 with an invariant curve. Invent. Math.211(2018), no. 1, 1–38

    2018

  9. [17]

    Algebraic Geom.27(2018), 497–551

    Dinh, Tien-Cuong; Sibony , Nessim: Density of positive closed currents, a theory of non-generic inter- sections.J. Algebraic Geom.27(2018), 497–551

    2018

  10. [18]

    Federer Herbert:Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, Band153Springer-Verlag New York Inc., New York, 1969

    1969

  11. [19]

    Ann.301 (1995), no

    Fornæss, John Erik; Sibony , Nessim: Oka’s inequality for currents and applications.Math. Ann.301 (1995), no. 3, 399–419

    1995

  12. [20]

    Fornæss, John Erik; Sibony , Nessim: Harmonic currents of finite energy and laminations.Geom. Funct. Anal.15(2005), no. 5, 962–1003. 40

    2005

  13. [21]

    Fornæss, John Erik; Sibony , Nessim: Riemann surface laminations with singularities.J. Geom. Anal. 18(2008), no. 2, 400–442

    2008

  14. [22]

    Fornæss, John Erik; Sibony , Nessim: Unique ergodicity of harmonic currents on singular foliations of P2.Geom. Funct. Anal.19(2010), no. 5, 1334–1377

    2010

  15. [23]

    (English summary)Adv

    Fu, Jixiang; Xiao, Jian: Relations between the K ¨ahler cone and the balanced cone of a K ¨ahler mani- fold. (English summary)Adv. Math.263(2014), 230–252

    2014

  16. [24]

    Ledrappier, Franc ¸ois; Young, Lai-Sang: The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin’s entropy formula.Ann. of Math. (2)122(1985), no. 3, 509–539

    1985

  17. [25]

    (French)Bull

    Lelong, Pierre: Int ´egration sur un ensemble analytique complexe. (French)Bull. Soc. Math. France 85(1957), 239–262

    1957

  18. [26]

    (French)Gordon & Breach, Paris-London-New York (Distributed by Dunod´editeur, Paris) 1968 ix+79 pp

    Lelong, Pierre: Fonctions plurisousharmoniques et formes diff ´erentielles positives. (French)Gordon & Breach, Paris-London-New York (Distributed by Dunod´editeur, Paris) 1968 ix+79 pp

    1968

  19. [27]

    (French)[Self-intersection inequalities for closed positive currents defined in projective varieties].Ann

    Meo, Michel: In ´egalit´es d’auto-intersection pour les courants positifs ferm´es d´efinis dans les vari´et´es projectives. (French)[Self-intersection inequalities for closed positive currents defined in projective varieties].Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)26(1998), n...

    1998

  20. [28]

    Nguy ˆen, Viˆet-Anh: Positive plurisubharmonic currents: Generalized Lelong numbers and Tangent theorems.arXiv:2111.11024.v2 [math.CV], 249 pages

  21. [29]

    Nguy ˆen, Viˆet-Anh: The generalized Lelong numbers and intersection theory .arXiv:2501.02150, 78 pages

  22. [30]

    (French) [Some extension problems for currents in complex analysis]Duke Math

    Sibony , Nessim: Quelques probl `emes de prolongement de courants en analyse complexe. (French) [Some extension problems for currents in complex analysis]Duke Math. J.52(1985), no. 1, 157– 197

    1985

  23. [31]

    Math.27(1974), 53-156

    Siu, Yum Tong: Analyticity of sets associated to Lelong numbers and the extension of closed positive currents.Invent. Math.27(1974), 53-156

    1974

  24. [32]

    (French) [Extension of closed, positive currents of finite mass]Invent

    Skoda, Henri: Prolongement des courants, positifs, ferm ´es de masse finie. (French) [Extension of closed, positive currents of finite mass]Invent. Math.66(1982), no. 3, 361–376

    1982

  25. [33]

    Second edition

    Sogge, Christopher D.: Fourier integrals in classical analysis. Second edition. Cambridge Tracts in Mathematics,210. Cambridge University Press, Cambridge, 2017. xiv+334 pp

    2017

  26. [34]

    Toma, Matei: A note on the cone of mobile curves.C. R. Math. Acad. Sci. Paris348(2010), no. 1-2, 71–73

    2010

  27. [35]

    I.Cambridge Studies in Advanced Math- ematics,76

    Voisin, Claire:Hodge theory and complex algebraic geometry. I.Cambridge Studies in Advanced Math- ematics,76. Cambridge University Press, Cambridge, 2007

    2007

  28. [36]

    Vigny , Gabriel: Lelong-Skoda transform for compact K¨ahler manifolds and self-intersection inequali- ties.(English summary)J. Geom. Anal.19(2009), no. 2, 433–451

    2009

  29. [37]

    With an appendix by S´ebastien Boucksom.J

    Witt Nystr ¨om, David: Duality between the pseudoeffective and the movable cone on a projective manifold. With an appendix by S´ebastien Boucksom.J. Amer. Math. Soc.32(2019), no. 3, 675–689. DEPARTMENT OFMATHEMATICS, NATIONALUNIVERSITY OFSINGAPORE, 10 LOWERKENTRIDGEROAD, SINGA...

    2019

Pith tools

Reviewed June 30, 2026 · model on record in the stance chip above.