Uniqueness of tangent currents for positive closed currents

Tuyen Trung Truong; Viet-Anh Nguyen

REVIEW 2 major objections 4 minor 1 cited by

Uniqueness of tangent currents for positive closed currents

T0 review · 2 major / 4 minor · reviewed 2026-08-08 · deepseek-v4-flash

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

Pith's one-line read Fast tube convergence pins a unique tangent current to a positive closed current along a submanifold.

desk verdict New uniqueness criterion for tangent currents that is worth refereeing; Section 6 has a repairable gap at the zero section and Theorem 1.13 depends on unpublished work. read the letter →

arxiv 2502.06532 v1 pith:KXZKDHIU submitted 2025-02-10 math.CV math.AG

Viet-Anh Nguyen , Tuyen Trung Truong This is my paper · ORCID

classification math.CVmath.AG MSC 32J2514C30

keywords positiveclosedcurrenttangentgeneralizedLelongnumberstronglyadmissiblemapuniquenesscriterioncomplexanalyticsetintersectiontheory

topics The Hodge Conjecture

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

This paper proves a sufficient condition under which a positive closed current has exactly one tangent current along a submanifold, not merely at a point. The condition is quantitative: the j-th average tube mass over a domain B must converge to the generalized Lelong number fast enough that the difference, divided by r, is integrable near r=0. When this holds, every rescaled family converges as |λ|→∞ to the same V-conic current, independent of the admissible map and metric. A local version checks the same integrability on a finite cover of B. For currents of integration over complex analytic sets the paper proves the required convergence rate is polynomial, so uniqueness holds there automatically.

What carries the argument

The proof is carried by the generalized Lelong numbers ν_j(T,B,r,τ,h), defined as normalized masses of τ_*T on tubes over B, together with strongly admissible maps τ that straighten a neighbourhood of B into the normal bundle E and are identity on V to first order. The main identity is the Lelong–Jensen formula for vector bundles (Theorem 2.8), which expresses the difference ν_j(r)-ν_j(r/2) as a vertical boundary term plus a local mass κ_j on the corona tube. A technique of admissible estimates compares κ_j with positive global mass indicators K_{j,q}; a key global inequality bounds these indicators, and two classical lemmas on functions on a punctured disk convert the resulting integrability bounds into convergence of the coefficients of the rescaled currents as λ→∞. The finite-cover version runs the same argument on each domain of a cover and matches the limits on overlaps.

What would settle it

To disprove the conditional claim, one would need a positive closed current in the stated approximability class for which both integrability conditions hold, yet two different tangent currents arise from two different strongly admissible maps. A concrete starting point is a product construction whose fibre factor is a direction-dependent tangent cone at a point, chosen so that its average-mean oscillation decays faster than the integrability thresholds while the tangent cone remains non-unique; comparing the limits for two maps would settle whether the theorem's hypotheses are sufficient.

Watch

Extended reading notes

Core claim

The central claim, stated as Theorem 1.9, is that uniqueness of tangent currents follows from an integrability condition on the generalized Lelong numbers. For a positive closed (p,p)-current T on a complex manifold X, a Kähler submanifold V of dimension l, and a relatively compact piecewise $C^{2}$ domain B⊂V, assume T=T^+-T^- near B with T^± in the class CL_{p;1,1}(B), meaning each is weakly approximable by smooth closed forms with uniformly bounded $C^{1}$ norms near the boundary of B. If for every j between max(0,l-p) and min(l,k-p) either the square-root integral of the difference between ν_j at radii r and r/2 or the plain integral of the difference between ν_j(r) and the limit ν_j is finite, then T admits a unique tangent current along B. The paper also establishes a finite-cover version with conditions on local mass indicators and proves that for the current of integration over a pure-codimension-p analytic set S the average means converge with rate O(r^ρ), so uniqueness holds there and the convergence speed is quantified.

Load-bearing premise

The proof rests on the assumption that, near the boundary of B, the current can be written as a difference of two positive closed currents that are limits of smooth forms with controlled size and controlled first derivatives; without that, the key formulas and mass bounds do not apply.

Editorial extensions

If this is right

For any positive closed current in the stated class satisfying either integrability condition, the rescaled family converges as λ→∞ rather than merely along subsequences, so the tangent current is a uniquely defined invariant of T along B.
The finite-cover theorem makes the criterion checkable locally: one only needs estimates on tube masses over pieces of a cover of B, which is useful when the singular support of T meets the boundary of B.
Every current of integration over a complex analytic set S whose intersection with B is relatively compact in B admits a unique tangent current along B, with the explicit rate O(r^ρ) for the approach of average means to their limits.
Because the generalized Lelong numbers are independent of the strongly admissible map, the uniqueness criterion is intrinsic even though the proof tracks approximations that are not holomorphic.
The result extends the classical point-based uniqueness criterion to positive-dimensional base sets, so it applies to excess-intersection situations where the expected intersection dimension is larger than the classical one.

Reading between the lines

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

The integrability conditions are sufficient, not necessary; a natural test is whether much slower, Dini-type decay of the tube averages still forces uniqueness, since the two key lemmas only need specific weighted integrability.
The equivalence between the κ_j conditions and the global mass indicators suggests a practical certificate: if one can bound κ_j by a small power of r on each piece of a cover, uniqueness follows without computing the generalized Lelong numbers.
For analytic sets, the polynomial rate O(r^ρ) should imply stability of the tangent current under holomorphic deformations of S, a consequence not stated in the paper.
The remark about weakening the Kähler assumption indicates that the true ingredients are the strongly admissible map and boundary C^1 control; testing the theorem on non-Kähler examples would clarify whether the integrability condition alone is the operative hypothesis.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear [Colleague],

The paper is a serious generalization of the Blel–Demailly–Mouzali uniqueness criterion from points to submanifolds V of positive dimension. That generalization was missing from the literature, and the conditions in Theorem 1.9 look like the right hypotheses: they are genuinely sufficient (no fitted constants, no invented objects) and reduce cleanly to global integrability via the admissible estimates in Section 4. The analytic-set result, Theorem 1.13, with the O(r^rho) rate, is a nice quantitative bonus. I agree with the reader's conditional accept.

The main soft spot is exactly where the stress-test points. Section 6 proves convergence of T_lambda only on E|B \ B, and the proof explicitly restricts to test forms supported away from the zero section. Tangent currents can be supported on B, and such support is not screened by the written argument. The stress-test's suggested counterexample [V] is excluded by the CL hypothesis (V is singular all the way to the boundary of B), but a point mass in the interior of B, e.g., T = ddc log(|z|^2+|w|^2) in C^2, satisfies the CL condition and has a tangent current supported on a single point of B; the restriction to E|B\B carries zero mass. So the gap is real. However, it is not load-bearing: because A_lambda fixes the base V, we have pi_* T_lambda = pi_* T for every lambda, hence any tangent current S satisfies pi_* S = pi_* T. Two cluster limits S_1 and S_2 agree on the complement, so D = S_1 - S_2 is V-conic and supported on B; V-conicity rules out normal derivatives, hence D = i_* R, and then R = pi_* D = 0. The authors should add this short step; as written it is missing.

The other soft spot is Theorem 1.13: it depends on the cylindrical cone construction in the in-progress [22]. That is a significant citation to unpublished work for a stated theorem; the referee should ask for an appendix or a public preprint version. The heavy use of [20] is acceptable but will slow verification.

Overall, the central argument holds up, the gap at the zero section is minor and repairable, and the result deserves referee time. I recommend a revision that fills in the zero-section argument and pins down the status of [22].

Referee Report

2 major / 4 minor

Summary. The paper claims a uniqueness criterion for tangent currents of positive closed currents along a Kähler submanifold. Theorem 1.9 states that if T = T^+ - T^- with T^± in the class CL_{p;1,1}(B), and if the averaged generalized Lelong numbers ν_j(T,B,r) converge to ν_j(T,B) fast enough so that the differences are integrable against r^{-1} dr, then the family (A_λ)_* τ_* T has a unique limit as |λ| → ∞. Theorem 1.10 gives a local version over a finite cover of B by piecewise C^2 domains. Theorem 1.13 shows that for currents of integration over complex analytic sets the differences ν_j(r) - ν_j have order O(r^ρ), so the criterion applies. The proofs reduce the integrability conditions to growth estimates of global mass indicators using Lelong–Jensen formulas, pass to local coefficient estimates, and then use the Blel–Demailly–Mouzali limiting argument for functions on the punctured disc.

Significance. If the main result is correct, it is a natural and valuable generalization of the Blel–Demailly–Mouzali criterion from points to higher-dimensional bases, with concrete applications to intersection theory. The paper does not fit free parameters and the analytic-set rate in Theorem 1.13 is a useful quantitative contribution. The arguments are not self-contained and rely heavily on the first author's long preprint [20], but the cited statements are precise enough to be checked. The main obstruction to accepting the paper as written is a gap in the final convergence argument near the zero section, which is exactly the region where tangent currents may carry mass.

major comments (2)

[Section 6, End of the proof of Theorem 1.9] The proof only establishes weak convergence of T_λ on E|B ∖ B, not on a neighborhood of B. The text explicitly says 'We only need to prove that Tλ converge weakly locally in E|B∖B' and then takes test functions supported in a relatively compact open set in E|B∖B. This is not sufficient: tangent currents are defined on all of π⁻¹(B) and may be supported on B. For example, if T = [V], then Remark 1.6 and Theorem 1.13 imply that the hypotheses of Theorem 1.9 hold, but T_λ = [V] for every λ; this current is zero on E|B∖B and equals [V] on B. Convergence away from B cannot distinguish [V] from 0. The same gap affects the proofs of Theorem 1.10 and the final statement of Theorem 1.13. The authors need either to prove weak convergence for test forms supported near B or to prove an extension principle, for instance that any two V-conic subsequential limits that agree on E|B∖B and have the same compactly supported cohomology class must coincide; neither argument is present.
[Proposition 4.7, proof of assertion (a)] After equation (4.10), the proof bounds the sum of K_{j,k-p-j}(T,B,r/2,r) by a constant times the sum of |κ_j(T,B,r/2,r)| plus c r^{1/4}, citing what appears to be Proposition 4.5 but should be Proposition 4.6(1). However, Proposition 4.6 is stated for T in the positive class CL_{p;1,1}(U,W), while the current T in Theorem 1.9 is a signed difference T^+ - T^- with T^± in that class. The displayed reduction therefore needs an explicit linearity argument: apply the estimates to T^+ and T^- separately, then use the Lelong–Jensen formula for each positive part to control κ_j(T^±) by differences of the corresponding ν_j(T^±,r). As written, the step is not justified and is load-bearing for the reduction of Theorem 1.9 to the mass-indicator integrability.

minor comments (4)

[Section 7, equation (7.3)] Equation (7.3) appears to be incomplete: it reads 'CpSq ∩ BverTubepB, 1q' with no asserted relation or equality. From the surrounding argument, the intended statement is presumably that this intersection is empty for r small enough, since this is what is used later to remove the first term in the formula for Mver(r1,r2). Please correct this.
[Proposition 4.7(a)] The reference to 'Proposition 4.5' in the proof of assertion (a) seems to be a typo; the inequality relating K_{j,k-p-j}(T,B,r/2,r) and |κ_j(T,B,r/2,r)| is Proposition 4.6(1).
[Throughout] There are minor grammatical and typographical issues, for example 'Let T be a positive closed currents' in the abstract, 'for for all' near (1.10), and 'τ is a admissible map' in Theorem 1.13. These do not affect the mathematics.
[References] Theorem 1.13 relies on the construction of cylindrical cones from the item marked 'In progress' in the references. Since this is a load-bearing external tool for that theorem, the authors should either provide a stable reference or include a self-contained statement of the needed construction.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the integrability hypotheses are not defined in terms of the uniqueness conclusion, and the heavy reliance on prior work is support-based rather than definitional.

full rationale

The paper's main implication (Theorems 1.9 and 1.10) is a genuine derivation: conditions (a) and (b) are integrability conditions on differences of generalized Lelong numbers, and the conclusion is that the dilated currents converge. The proof reduces these conditions to decay of mass indicators, then applies the Blel-Demailly-Mouzali limiting lemma; no fitted parameter is renamed as a prediction, and no conclusion is built into an assumption. The paper does lean substantially on the first author's prior work [20] for the CL_{p;1,1} classes, the Lelong-Jensen formula (Theorem 2.8), the mass estimate (Theorem 4.2), and the admissible-estimates technique. This is heavy self-citation, but the cited results are prior mathematical theorems with stated assumptions that do not include the uniqueness conclusion, so the new implication is not forced by the citation alone. Theorem 1.13 uses the cylindrical cone construction from [22], which is listed as unpublished and by the same first author; this is a legitimate concern about support and completeness, but not circularity, because the construction is not assumed to prove the theorem. The skeptic's objection that Section 6 proves convergence only on E|B \ B and does not directly control tangent currents supported on the zero section is a possible gap in the proof of uniqueness, not a circular step: it concerns whether the argument establishes the stated conclusion, not whether the conclusion is equivalent to the input by construction. Overall, no circular step can be exhibited from the paper's own equations.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper imports most of its working machinery from the first author's prior work, especially [20] for Lelong numbers, tangent existence, Lelong-Jensen formulas, and mass estimates, and [22] for cylindrical cones. No numerical parameters are fitted to data. The key assumptions are the approximability class CL and the existence of strongly admissible maps.

assumptions (7)

domain assumption Lelong-Jensen formula for approximable currents (Theorem 2.8) holds for T in the class CL_{p;1,1}(B).
Imported from [20, Theorem 4.5]; used to relate average means to tangential integrals in Section 4.
domain assumption Uniform mass bound K_{j,q}(T,r) < c for currents in \tilde{CL}_{p;1,1}(U,W) (Theorem 4.2).
Imported from [20, Theorem 8.7]; underpins Propositions 4.3 and 4.5.
domain assumption Existence of strongly admissible maps along B for X Kähler (Proposition 2.7) and along each domain B_i.
Needed to define the coordinate models and the currents T#_r; cited from [20] and Dinh-Sibony.
domain assumption The class CL_{p;1,1}(B) contains integration currents over analytic sets (Remark 1.6).
Makes Theorem 1.13 applicable via Theorem 1.9.
domain assumption Theorem 1.7 (existence of generalized Lelong numbers and tangent currents for approximable T) holds.
Imported from [20, Theorem 1.11]; used in Propositions 4.6 and 4.7 to identify limits and intrinsicness.
standard math Stokes' theorem and Sard's theorem apply to boundaries of tubes and analytic sets.
Used in Section 7 to pass integrals over A_{r^{-1}}(τ(S)) to boundary integrals and to control critical values.
standard math The classical BDM limiting lemmas (6.2, 6.3) and volume estimate (7.1) are valid.
Used in Section 6 to pass from estimates on derivatives to existence of limits, and in Section 7 to get polynomial volume growth.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Uniqueness of tangent currents for positive closed currents." pith.science (2026). https://pith.science/paper/KXZKDHIU

@misc{pith2026250206532,
  author       = {Pith},
  title        = {Pith review of: Uniqueness of tangent currents for positive closed currents},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KXZKDHIU}},
  note         = {Machine review of arXiv:2502.06532}
}

abstract

Let $X$ be a complex manifold $X$ of dimension $k,$ and let $V\subset X$ be a K\"ahler submanifold of dimension $l,$ and let $B\subset V$ be a piecewise $\mathcal{C}^2$-smooth domain. Let $T$ be a positive closed currents of bidegree $(p,p)$ in $X$ such that $T$ satisfies a mild reasonable assumption in a neighborhood of $\partial B$ in $X$ and that the $j$-th average mean $\nu_j(T,B,r)$ for every $j$ with $\max(0,l-p)\leq j\leq\min(l,k-p)$ converges sufficiently fast to the $j$-th generalized Lelong number $\nu_j(T,B)$ as $r$ tends to $0$ so that $r^{-1}(\nu_j(T, B,r)-\nu_j( T,B))$ is locally integrable near $r=0.$ Then we show that $T$ admits a unique tangent current along $B.$ A local version where we replace the condition of $T$ near $B$ by the conditions on a finite cover of $B$ by piecewise $\mathcal{C}^2$-smooth domains in $V$ is also given. When $T$ is a current of integration over a complex analytic set, we show that $\nu_j(T,B,r)-\nu_j(T,B)=O(r^\rho)$ for some $\rho>0,$ and hence this condition is satisfied. Our result may be viewed as a natural generalization of Blel-Demailly-Mouzali's criterion from the case $l=0$ to the case $l>0.$ The result has applications in the intersection theory of positive closed currents.

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

Continuous local potential functionals and the Dinh-Sibony product
math.CV 2026-07 conditional novelty 6.0 of 10

On any complex manifold, the Dinh-Sibony product of three positive closed currents is well defined and associative when the first current has continuous local potential functionals and the other two satisfy Condition (I).

Reference graph

Works this paper leans on

26 extracted references · 24 canonical work pages · cited by 1 Pith paper

[22]

In progress

Nguyˆ en, Viˆ et-Anh: Thie type theorems and the general ized Lelong numbers for complex analytic sets. In progress

work page
[20]

arXiv:2111.11024v2, 241 pages

Nguyˆ en, Viˆ et-Anh: Positive plurisubharmonic curre nts: Generalized Lelong numbers and Tangent theorems. arXiv:2111.11024v2, 241 pages

arXiv
[1]

Results Math

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

work page 1996
[2]

(French) [Tangent cone of a nonnegative closed p1, 1q current] C

Blel, Mongi: Cˆ one tangent ` a un courant positif ferm´ e d e type p1, 1q. (French) [Tangent cone of a nonnegative closed p1, 1q current] C. R. Acad. Sci. Paris S´ er . I Math.309 (1989), no. 8, 543-546

work page 1989
[3]

(French) [Existence of the tangent cone of a closed positive current] Ark

Blel, Mongi; Demailly , Jean-Pierre; Mouzali, Mokhtar: Sur l’existence du cˆ one tangent ` a un courant positif ferm´ e. (French) [Existence of the tangent cone of a closed positive current] Ark. Mat. 28 (1990), no. 2, 231–248

work page 1990
[4]

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

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

work page 2012
[5]

Dinh, Tien-Cuong; Nguyˆ en, Viˆ et-Anh; Vu, Duc-Viet: Super-potentials, densities of currents and num- ber of periodic points for holomorphic maps. Adv. Math. 331 (2018), 874–907

work page 2018
[6]

Indiana Univ

Dinh, Tien-Cuong; Nguyˆ en, Viˆ et-Anh; Truong, T uyen Trung: Equidistribution for meromorphic maps with dominant topological degree. Indiana Univ. Math. J. 64 (2015), no. 6, 1805–1828

work page 2015

Show all 26 references

[7]

Dinh, Tien-Cuong; Nguyˆ en, Viˆ et-Anh; Truong, T uyen Trung: Growth of the number of periodic points for meromorphic maps. Bull. Lond. Math. Soc. 49 (2017), no. 6, 947–964

2017
[8]

Dinh, Tien-Cuong; Sibony , Nessim: Regularization of cu rrents and entropy .Ann. Sci. ´Ecole Norm. Sup. (4) 37 (2004), no

2004
[9]

Dinh, Tien-Cuong; Sibony , Nessim: Equidistribution of saddle periodic points for H´ enon-type auto- morphisms of Ck. Math. Ann. 366 (2016), no. 3-4, 1207–1251

2016
[10]

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
[11]

Dinh, Tien-Cuong; Sibony , Nessim: Unique ergodicity f or foliations in P2 with an invariant curve. Invent. Math. 211 (2018), no. 1, 1–38

2018
[12]

Second edition

Fulton, William: Intersection theory . Second edition . Ergebnisse der Mathematik und ihrer Gren- zgebiete. 3. Folge. A Series of Modern Surveys in Mathematic s [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics ], vol. 2. Springer...

1998
[13]

Pure and Applied Mathematics

Grifﬁths, Phillip; Harris, Joseph: Principles of alge braic geometry . Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York, 1978. xii+813 pp

1978
[14]

Several complex variables (Proc

Harvey , Reese: Holomorphic chains and their boundarie s. Several complex variables (Proc. Sympos. Pure Math., Vol. XXX, Part 1, Williams Coll., Williamstown, M ass., 1975), pp. 309–382. Amer . Math. Soc., Providence, R. I., 1977

1975
[15]

Internat

Kaufmann, Lucas: Self-intersection of foliation cycl es on complex manifolds. Internat. J. Math. 28 (2017), no. 8, 1750054, 18 pp

2017
[16]

Kaufmann, Lucas; Vu, Duc-Viet: Density and intersecti on of p1, 1q-currents. J. Funct. Anal. 277 (2019), no. 2, 392–417

2019
[17]

International Symposium in Memory of Hua Loo Keng, Vol

Kiselman, Christer O.: Tangents of plurisubharmonic f unctions. International Symposium in Memory of Hua Loo Keng, Vol. II (Beijing, 1988), 157–167, Springer , Berlin,1991

1988
[18]

(French) Bull

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

1957
[19]

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

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

1968
[21]

arXiv:2501.02150 [math.CV ,math.AG, math.DG, math.DS] , 78 pages

Nguyˆ en, Viˆ et-Anh: The generalized Lelong numbers an d intersection theory . arXiv:2501.02150 [math.CV ,math.AG, math.DG, math.DS] , 78 pages

arXiv
[23]

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]

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

Skoda, Henri: Prolongement des courants, positifs, fe rm´ es de masse ﬁnie. (French) [Extension of closed, positive currents of ﬁnite mass] Invent. Math. 66 (1982), no. 3, 361–376. UNIQUENESS OF T ANGENT CURRENTS 33

1982
[25]

Thie, Paul R.: The Lelong number of a point of a complex an alytic set. Math. Ann. 172 (1967), 269–312

1967
[26]

Vu, Duc-Viet: Densities of currents on non-K¨ ahler manifolds. Int. Math. Res. Not. IMRN. 2021, no. 17, 13282–13304. UNIVERSIT ´E DE LILLE , L ABORATOIRE DE MATH ´EMATIQUES PAUL PAINLEV ´E, CNRS U.M.R. 8524, 59655 VILLENEUVE D ’ASCQ CEDEX , F RANCE . AND VIETNAM INSTITUTE FOR ...

2021

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.

[1] [22]

In progress

Nguyˆ en, Viˆ et-Anh: Thie type theorems and the general ized Lelong numbers for complex analytic sets. In progress

work page

[2] [20]

arXiv:2111.11024v2, 241 pages

Nguyˆ en, Viˆ et-Anh: Positive plurisubharmonic curre nts: Generalized Lelong numbers and Tangent theorems. arXiv:2111.11024v2, 241 pages

arXiv

[3] [1]

Results Math

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

work page 1996

[4] [2]

(French) [Tangent cone of a nonnegative closed p1, 1q current] C

Blel, Mongi: Cˆ one tangent ` a un courant positif ferm´ e d e type p1, 1q. (French) [Tangent cone of a nonnegative closed p1, 1q current] C. R. Acad. Sci. Paris S´ er . I Math.309 (1989), no. 8, 543-546

work page 1989

[5] [3]

(French) [Existence of the tangent cone of a closed positive current] Ark

Blel, Mongi; Demailly , Jean-Pierre; Mouzali, Mokhtar: Sur l’existence du cˆ one tangent ` a un courant positif ferm´ e. (French) [Existence of the tangent cone of a closed positive current] Ark. Mat. 28 (1990), no. 2, 231–248

work page 1990

[6] [4]

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

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

work page 2012

[7] [5]

Dinh, Tien-Cuong; Nguyˆ en, Viˆ et-Anh; Vu, Duc-Viet: Super-potentials, densities of currents and num- ber of periodic points for holomorphic maps. Adv. Math. 331 (2018), 874–907

work page 2018

[8] [6]

Indiana Univ

Dinh, Tien-Cuong; Nguyˆ en, Viˆ et-Anh; Truong, T uyen Trung: Equidistribution for meromorphic maps with dominant topological degree. Indiana Univ. Math. J. 64 (2015), no. 6, 1805–1828

work page 2015