Holomorphic Foliation Theory — Problem 1

Definitions and mathematical statement

Let $\mathbb D:=\{z\in{\mathbb{C}}:|z|<1\}$ and $\mathbb D(r):=\{z\in{\mathbb{C}}:|z|<r\}$ for $0<r<1$. The supplied excerpt proposes replacing a weighted current $\tau_r^a$ by the unweighted current, called an Ahlfors current, \[ A(r):=\int_{\mathbb D(r)}\phi_a^*(\omega), \qquad \sigma_r^a:=\frac{1}{A(r)}(\phi_a)_*[\mathbb D(r)]. \] Here $[\mathbb D(r)]$ denotes integration over the disc. The notation $A(r)$ is retained from the source; its defining expression also involves $a$. Whenever the maps, forms, and integrals are defined and $0<A(r)<\infty$, the pushforward has the action \[ \langle\sigma_r^a,\eta\rangle =\frac{\displaystyle\int_{\mathbb D(r)}\phi_a^*(\eta)} {\displaystyle\int_{\mathbb D(r)}\phi_a^*(\omega)} \] on smooth compactly supported test forms $\eta$ of bidegree $(1,1)$ on the ambient complex manifold.

Question (Dinh--Sibony, Problem 1, 2018). With the data and hypotheses intended in the source, does \[ \sigma_r^a\longrightarrow T\qquad\text{as }r\uparrow1 \quad\text{for }T\text{-almost every }a? \] Under the usual interpretation as weak convergence of currents, this asks whether, for $T$-almost every $a$, every such test form $\eta$ satisfies \[ \lim_{r\uparrow1} \frac{\displaystyle\int_{\mathbb D(r)}\phi_a^*(\eta)} {\displaystyle\int_{\mathbb D(r)}\phi_a^*(\omega)} =\langle T,\eta\rangle. \] Unresolved qualifications. The supplied excerpt does not specify the ambient manifold, the foliation, the admissible points $a$, the maps $\phi_a$, the form $\omega$, or the hypotheses and normalization imposed on $T$. It also does not define the measure meant by $T$-almost every point. The positivity and finiteness needed for the displayed normalization are not stated. Thus this is a transcription and conditional interpretation of the question, not a fully specified conjecture with recoverable hypotheses. The limit is along all radii approaching $1$, as written in the supplied question.

Short English statement

Dinh and Sibony ask whether the normalized unweighted integration currents of the discs $\phi_a(\mathbb D(r))$ converge to $T$ as $r\uparrow1$, for $T$-almost every $a$. The excerpt omits the surrounding geometric hypotheses and the meaning of this almost-everywhere condition. The registry labels the problem open; that label is supplied curation, not an independently verified status assessment.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $800674$, exact source code catalogue entry, title "Holomorphic Foliation Theory --- Problem 1." The record attributes the problem to Dinh and Sibony, with proposed year 2018, and labels its status open.
• Dinh and Sibony, Some Open Problems on Holomorphic Foliation Theory (2018), Problem 1. Supplied source URL: https://arxiv.org/abs/1807.05640.
• Supplied provenance records access on 2026-10-16 and extraction by arxiv_latex. These are source-record provenance statements; the source was not independently accessed or verified for this draft.