Siu's analyticity theorem for positive pluriharmonic currents
Abstract
Let T be a positive ddc-closed current of bidimension (q,q) on a compact Kähler manifold X. For every c>0, let Ec(T) be the set of points of X where the Lelong number of T is larger or equal to c. We show that Ec(T) is an analytic subset of dimension at most q of X. Moreover, the following Siu decomposition holds T=∑i∈ I λi[Vi] +T0, where {Vi}i∈ I is a (possibly empty) finite or countable family of compact analytic subsets of dimension q in X, λi∈ℝ+, and T0 is a positive ddc-closed current such that Ec(T0) is an analytic subset of dimension at most q−1 of X for every c>0. The proof relies on the duality between the pseudoeffective cone and the movable cone of a compact Kähler manifold, recently obtained by Tosatti (2026), together with a theorem of Vigny (2009) and the theory of density currents for positive ddc-closed currents developed by Sibony, the first and third authors.
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BibTeX
@misc{dinh2026sius,
title = {{Siu's analyticity theorem for positive pluriharmonic currents}},
author = {Tien-Cuong Dinh and Duc-Bao Nguyen and Viet-Anh Nguyen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.29680},
eprint = {2606.29680},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.29680},
}
RIS
TY - UNPB
TI - Siu's analyticity theorem for positive pluriharmonic currents
AU - Tien-Cuong Dinh
AU - Duc-Bao Nguyen
AU - Viet-Anh Nguyen
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2606.29680
UR - https://arxiv.org/abs/2606.29680
AB - Let T be a positive dd^c-closed current of bidimension (q,q) on a compact Kähler manifold X. For every c>0, let E_c(T) be the set of points of X where the Lelong number of T is larger or equal to c. We show that E_c(T) is an analytic subset of dimension at most q of X. Moreover, the following Siu decomposition holds T=∑_i∈ I λ_i[V_i] +T_0, where {V_i}_i∈ I is a (possibly empty) finite or countable family of compact analytic subsets of dimension q in X, λ_i∈ℝ^+, and T_0 is a positive dd^c-closed current such that E_c(T_0) is an analytic subset of dimension at most q−1 of X for every c>0. The proof relies on the duality between the pseudoeffective cone and the movable cone of a compact Kähler manifold, recently obtained by Tosatti (2026), together with a theorem of Vigny (2009) and the theory of density currents for positive dd^c-closed currents developed by Sibony, the first and third authors.
ER -