Answers to three problems in Dinh-Sibony's list on holomorphic foliation theory
Abstract
We address Problems~2.2(i), 2.2(ii), and~2.4 in Dinh--Sibony's list. For a saturated holomorphic curve foliation on a compact Kähler manifold with finitely many singularities, every positive directed ddc-closed current has finite Poincaré mass. Thus linearizability is not needed for the geometric ergodic theorem in this setting. We construct a projective surface with dense hyperbolic leaves and a diffuse extremal current whose Nevanlinna averages converge at a uniform reciprocal logarithmic rate. This is an existence result for a chosen foliation, not a rate theorem for an arbitrary prescribed foliation. Under the hypotheses of the first result, every hyperbolic leaf has a logarithmic lower bound for its Nevanlinna characteristic, even if no diffuse directed current exists. Positive-dimensional singular sets are not treated.
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BibTeX
@misc{chen2026answers,
title = {{Answers to three problems in Dinh-Sibony's list on holomorphic foliation theory}},
author = {Zhangchi Chen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00228},
eprint = {2610.00228},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00228},
}
RIS
TY - UNPB
TI - Answers to three problems in Dinh-Sibony's list on holomorphic foliation theory
AU - Zhangchi Chen
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00228
UR - https://arxiv.org/abs/2610.00228
AB - We address Problems~2.2(i), 2.2(ii), and~2.4 in Dinh--Sibony's list. For a saturated holomorphic curve foliation on a compact Kähler manifold with finitely many singularities, every positive directed dd^c-closed current has finite Poincaré mass. Thus linearizability is not needed for the geometric ergodic theorem in this setting. We construct a projective surface with dense hyperbolic leaves and a diffuse extremal current whose Nevanlinna averages converge at a uniform reciprocal logarithmic rate. This is an existence result for a chosen foliation, not a rate theorem for an arbitrary prescribed foliation. Under the hypotheses of the first result, every hyperbolic leaf has a logarithmic lower bound for its Nevanlinna characteristic, even if no diffuse directed current exists. Positive-dimensional singular sets are not treated.
ER -