Smooth Levi-Flat Hypersurfaces in Complex Projective Planes

Definitions and mathematical statement

Let $\mathbb{CP}^{2}=({\mathbb{C}}^{3}\setminus\{0\})/{\mathbb{C}}^{\times}$ be the complex projective plane, with its standard complex structure $J$. For a $C^{\infty}$ embedded real hypersurface $M\subset\mathbb{CP}^{2}$ without boundary, define its complex tangent distribution by \[ H_pM=T_pM\cap J(T_pM),\qquad p\in M. \] This distribution has real rank two. The hypersurface is Levi-flat if $HM$ is involutive: for every open subset $U\subset M$ and all smooth vector fields $X,Y$ on $U$ taking values in $HM$, the bracket $[X,Y]$ also takes values in $HM$. Equivalently, the complex tangent distribution defines a foliation by complex curves.

The supplied question is: does $\mathbb{CP}^{2}$ contain a smooth Levi-flat real hypersurface?

A precise candidate for the intended global question is whether there exists a nonempty, closed, $C^{\infty}$ embedded real three-dimensional submanifold $M\subset\mathbb{CP}^{2}$, without boundary, whose complex tangent distribution is involutive. Here closed means closed as a subset of $\mathbb{CP}^{2}$, so such an $M$ is compact.

Unresolved qualification. The supplied statement does not explicitly require closedness or compactness, nor does it specify boundary conditions. Thus the closed, boundaryless formulation above is a candidate interpretation, not an additional hypothesis established by the record. The intended global scope must be checked before this draft is accepted.

Short English statement

The question asks whether the complex projective plane contains a smooth real hypersurface foliated by complex curves. The intended requirement that the hypersurface be closed and without boundary is not explicit in the supplied statement and remains unresolved. The supplied record labels the problem open.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003930$, in the Oberwolfach Reports Open Problems collection; exact source problem code catalogue entry. The record lists proposed year $2018$ and supplies no named proposer.
• Supplied report citation: Geometric Methods of Complex Analysis (2019), Oberwolfach Reports, pp. 2253--2302, DOI 10.4171/owr/2018/37. https://doi.org/10.4171/owr/2018/37. The supplied background attributes the problem to workshop open-problem discussions documented in Oberwolfach Reports.
• Supplied literature source: Oka Domains in Euclidean Spaces. https://academic.oup.com/imrn/article/2024/3/1801/7046029.
• Supplied literature source: Levi-flats in CP$^{n}$: a survey. https://math.sci.uwo.ca/~shafikov/papers/levi-flats2025.pdf.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and reports nonexistence in complex projective dimension at least three and for real-analytic hypersurfaces in $\mathbb{CP}^{2}$, while describing the smooth $\mathbb{CP}^{2}$ question as longstanding and open. This is supplied dated curation, not an independently verified literature assessment. Its precise hypotheses, including global closedness conditions, require checking against the cited sources.
• The registry labels the statement corrected_verified and says that notation, typography, and sentence structure were repaired while preserving mathematical content. This is supplied editorial metadata, not independent mathematical verification.