Holomorphic Foliation Theory — Problem 6

Definitions and mathematical statement

Let $X={\mathbb{P}}^2({\mathbb{C}})$ be the complex projective plane, with complex structure $J$. For a real hypersurface $M\subset X$ of class at least $C^2$, its complex tangent distribution is \[ H_pM=T_pM\cap J(T_pM),\qquad p\in M. \] This distribution has real rank two. The hypersurface is Levi-flat if $HM$ is integrable; equivalently, its Levi form vanishes identically. Locally, if $M=\{\rho=0\}$, where $\rho$ is a real defining function with ${\,\mathrm{d}}\rho\ne0$ on $M$, this means \[ (\partial\bar\partial\rho)_p(v,\bar v)=0 \quad\text{for every }p\in M\text{ and }v\in T_p^{1,0}M. \] The resulting local leaves are complex curves.

The explicit question in the supplied record is whether there exists a nonempty Levi-flat real hypersurface \[ M\subset{\mathbb{P}}^2({\mathbb{C}}) \] of smooth regularity, and whether such a hypersurface exists of real analytic regularity. Here smooth is conventionally read as $C^\infty$, although the source record does not specify a regularity threshold.

The global conditions on $M$ are unresolved in the supplied text: it does not explicitly require $M$ to be closed in ${\mathbb{P}}^2({\mathbb{C}})$, compact, or without boundary, and it does not explicitly distinguish embedded from immersed hypersurfaces. In particular, containment in the compact ambient space does not by itself impose compactness on $M$. These conditions materially affect the intended existence question and must be checked before this becomes a precise global formulation; no compactness or boundary hypothesis is silently added here.

The supplied preliminary discussion concerns a foliation $\mathscr F$ and a leaf $L$ that are not introduced in the excerpt. It describes an old question attributed to Camacho--Lins Neto--Sad, with references [CLS] and [FS3], and reports a positive answer when $\mathscr F$ admits a directed positive closed $(1,1)$-current $T$, in particular when it has an invariant algebraic curve. It states that $\operatorname{supp}T$ must contain a singularity, since otherwise $\{T\}^2=0$, which the discussion calls impossible in ${\mathbb{P}}^2$. It further describes constructing a directed positive $dd^c$-closed current $S$, supported by $\overline L$, and argues that $S$ and $T$ must intersect, with their intersection expected at singularities because both currents are directed by the foliation. The theory of densities for currents and [DS1] are suggested for this purpose. The question receiving this purported positive answer, the hypotheses on $\mathscr F$ and $L$, and the connection of this discussion to the explicit hypersurface question are not supplied. These remarks are source context, not additional established conclusions or a complete auxiliary statement.

Short English statement

The explicit question asks whether the complex projective plane contains a smooth Levi-flat real hypersurface, and whether one can be real analytic. Levi-flatness means that the complex tangent planes integrate locally to complex curves. The record omits essential global conditions, including closedness, compactness, and boundary conditions, so the intended global existence problem requires clarification. Its supplied status is open; this is a registry label, not an independently verified literature assessment.

Sources

• Local UnsolvedMath registry, permanent ID $800679$, exact source problem code catalogue entry, title Holomorphic Foliation Theory --- Problem 6, in the AMR Open Problem Lists collection. The supplied record attributes the problem to Dinh and Sibony, with proposed year 2018, and labels it open.
• Dinh and Sibony, Some Open Problems on Holomorphic Foliation Theory (2018), Problem 6. Supplied source URL: https://arxiv.org/abs/1807.05640. The background records extraction by arxiv_latex and an access date of 2026-10-16. These are supplied provenance details; the source has not been independently accessed or verified for this draft.
• The preliminary discussion attributes an old question to Camacho--Lins Neto--Sad under the reference label [CLS], and also cites [FS3]. Complete bibliographic entries and the precise scope of this attribution are absent from the supplied record.
• The preliminary discussion cites [DS1] concerning the theory of densities for currents. No complete bibliographic entry or separate link is supplied.