Siu's analyticity theorem for positive pluriharmonic currents

Tien-Cuong Dinh; Viet-Anh Nguyen

arxiv: 2606.29680 · v1 · pith:WNG6FGK6new · submitted 2026-06-29 · 🧮 math.CV · math.AG

Siu's analyticity theorem for positive pluriharmonic currents

Tien-Cuong Dinh , Viet-Anh Nguyen This is my paper

Pith reviewed 2026-06-30 04:28 UTC · model grok-4.3

classification 🧮 math.CV math.AG

keywords positive currentsLelong numbersanalytic setsSiu decompositionprojective manifoldsddc-closed currentseffective conecohomology classes

0 comments

The pith

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

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

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}.

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

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.

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.

Where Pith is reading between the lines

These are 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.

Load-bearing premise

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

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.

read the original 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 $H^{1,1}(X,\mathbb{R}).$

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

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

The main result is that on a projective manifold, the superlevel sets of the Lelong number for a positive ddc-closed bidimension-(1,1) current are analytic of dimension at most 1, and the current admits a Siu-type decomposition into a sum of currents of integration along analytic curves plus a remainder with Lelong numbers vanishing off a countable set. The abstract also records the consequence that such currents without mass on proper analytic sets have cohomology classes in the dual of the effective cone.

What is new is the passage from the usual closed case to the ddc-closed (pluriharmonic) case while keeping the analyticity conclusion, plus the explicit link to the effective cone. The statement is precise about the required hypotheses: projectivity of the ambient space and bidimension (1,1). That matches the weakest assumption listed and avoids overclaiming.

The limitation is that only the abstract is visible here, so the actual proof steps, any use of projectivity, and the handling of the remainder term cannot be checked. If the argument turns out to rely on some reduction that works only in this dimension or only after passing to a resolution, that would need to be spelled out. No circularity is visible at the level of the claim.

This is a specialized note in complex geometry. Readers already working with positive currents, Lelong numbers, and the effective cone on projective varieties will find the statement useful; others can skip it. The result is narrow enough that it does not need to be groundbreaking to be worth refereeing.

I would send it to peer review. The claim is concrete, the hypotheses are stated up front, and the consequence is stated cleanly. A referee can check whether the proof actually delivers the analyticity and the decomposition without extra assumptions.

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

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.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 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.

axioms (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.

pith-pipeline@v0.9.1-grok · 5755 in / 1514 out tokens · 66020 ms · 2026-06-30T04:28:29.455783+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

37 extracted references · 2 canonical work pages

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

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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
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(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

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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

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(French)Bull

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

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(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

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(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), no. 1, 161–184

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

work page arXiv
[29]

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

work page arXiv
[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
[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
[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
[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
[34]

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

2010
[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
[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
[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, SINGAPORE119076.http://www.math.nus.edu.sg/„matdtc Email address:matdtc@nus.edu.sg UN...

2019

[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

[9] [9]

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

2023

[10] [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

[11] [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

[12] [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

[13] [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

[14] [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

[15] [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

[16] [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

[17] [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

[18] [18]

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

1969

[19] [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

[20] [20]

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

2005

[21] [21]

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

2008

[22] [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

[23] [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

[24] [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

[25] [25]

(French)Bull

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

1957

[26] [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

[27] [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), no. 1, 161–184

1998

[28] [28]

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

work page arXiv

[29] [29]

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

work page arXiv

[30] [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

[31] [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

[32] [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

[33] [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

[34] [34]

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

2010

[35] [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

[36] [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

[37] [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, SINGAPORE119076.http://www.math.nus.edu.sg/„matdtc Email address:matdtc@nus.edu.sg UN...

2019