Nonexistence of Homogeneous Levi-Flat Hypersurfaces in ${\mathbb{C}\mathbb{P}}^2$

Abstract

We investigate the longstanding question of whether compact Levi-flat hypersurfaces exist in the complex projective plane ${\mathbb{CP}}^2$. While the nonexistence of closed real-analytic Levi-flat hypersurfaces in ${\mathbb{CP}}^n$ for $n>2$ is well known, the case $n=2$ remains open. By combining techniques from the classification of homogeneous CR-manifolds with projective foliation geometry, we prove that no homogeneous Levi-flat hypersurfaces exist in ${\mathbb{CP}}^2$, thus partially resolving the problem under natural symmetry assumptions.

1. Introduction

The question of whether closed Levi-flat hypersurfaces exist in complex projective spaces is one of the most intriguing and longstanding problems in several complex variables and complex geometry. A Levi-flat hypersurface is a real hypersurface whose Levi form vanishes identically, equivalently admitting a foliation by complex hypersurfaces. The existence or nonexistence of such hypersurfaces in compact Kähler manifolds, such as projective spaces, connects complex foliation theory, differential geometry, and CR geometry.

A recent survey by Shafikov [1] provides a comprehensive account of the current state of knowledge on Levi-flat hypersurfaces in projective spaces. For complex projective spaces ${\mathbb{CP}}^n$ with $n>2$, it is now classical that no closed real-analytic Levi-flat hypersurfaces exist, with several independent approaches establishing this fact. One method extends the Levi foliation holomorphically to a global (singular) codimension-one foliation on ${\mathbb{CP}}^n$ and then derives a contradiction from the geometry of its complementary domains [2]. The real-analytic Levi-flat condition concretely produces local holomorphic first integrals: for each $p\in M$ there is a neighborhood $U\subset {\mathbb{CP}}^n$ and a holomorphic function $F_U$ such that

$$M\cap U=\{Im{F}_U=0\},\mathrm{with}\mathrm{leaves}\{F_U=\mathrm{const}\}.$$

Analytic continuation and compatibility on overlaps glue these data into a meromorphic codimension-one holomorphic foliation $\mathcal{F}$ on ${\mathbb{CP}}^n$, equivalently given by a meromorphic 1-form $\omega$ with

$$\omega \wedge d\omega =0,\mathrm{Sing}\left(\mathcal{F}\right)\subset \{\omega =0\},$$

whose singular set has a complex codimension of at least 2. Locally the two components of ${\mathbb{CP}}^n\setminus M$ are

$${\Omega }_{\pm }\cap U=\{\pm Im{F}_U>0\},$$

and are saturated by $\mathcal{F}$ leaves. Patching the local potentials $\pm Im{F}_U$ (or using a leafwise-harmonic potential composed with a transverse distance to $\mathbb{R}$) yields global plurisubharmonic exhaustions on ${\Omega }_{\pm }$; for $n\ge 3$ the Levi problem and Andreotti–Grauert theory imply that each ${\Omega }_{\pm }$ is Stein. Lins Neto proves that such Stein complementary domains cannot occur for a real-analytic Levi-flat $M\subset {\mathbb{CP}}^n$ invariant under a global codimension-one holomorphic foliation when $n\ge 3$, yielding the desired contradiction.

A second, topological route uses Haefliger’s theory of codimension-one transversely holomorphic foliations [3]. The extended foliation determines a classifying map

$$f:{\mathbb{CP}}^n⇢B{\Gamma }_{\mathrm{hol}}^1,$$

and the holomorphic normal line bundle $N_{\mathcal{F}}$ is the pullback of the universal transverse bundle $U\to B{\Gamma }_{\mathrm{hol}}^1$. Hence its Chern classes are pullbacks of universal Haefliger classes:

$$c_1\left(N_{\mathcal{F}}\right)=f^{*}\left({\kappa }_1\right)\in H^2({\mathbb{CP}}^n;\mathbb{Z}).$$

Comparing with the ambient Chern classes via the exact sequence, on a general line $\mathcal{l}\simeq {\mathbb{P}}^1$ transverse to $\mathcal{F}$ we have

$${0⟶T\mathcal{F}|}_{\mathcal{l}}⟶T{\mathbb{CP}}^n{{|}_{\mathcal{l}}⟶N_{\mathcal{F}}|}_{\mathcal{l}}⟶0,$$

and the splitting

$$T{\mathbb{CP}}^n{{|}_{\mathcal{l}}\simeq \mathcal{O}\left(2\right)\oplus \mathcal{O}{\left(1\right)}^{\oplus (n-1)},T\mathcal{F}|}_{\mathcal{l}}\simeq \mathcal{O}{\left(1\right)}^{\oplus (n-1)}.$$

Hence $deg{N}_{\mathcal{F}}{|}_{\mathcal{l}}=(n+1)-(n-1)=2>0$. This follows from the positivity of the Fubini–Study curvature and transversality. On the other hand, the Haefliger class $f^{*}\left({\kappa }_1\right)=c_1\left(N_{\mathcal{F}}\right)$ is constrained by the transverse holomorphic structure and holonomy via the classifying map, leading to conditions incompatible with this positivity. This characteristic-class contradiction rules out such Levi-flat hypersurfaces for $n>2$.

However, as emphasized in [1], these nonexistence arguments for $n\ge 3$ break down in the special case $n=2$ for distinct, dimension-specific reasons. In the Lins Neto extension method, one extends the Levi foliation across the Stein complements to a global foliation on ${\mathbb{CP}}^n$; when $n>2$, the extended foliation necessarily has a positive-dimensional (codimension-two) singular set, forcing a contradiction, whereas in ${\mathbb{CP}}^2$ the singularities may be isolated points, and no such contradiction arises. In Siu’s method, a crucial step uses the vanishing

$$H_{\mathrm{cpt}}^2(X,\mathcal{O})=0$$

for a Stein component X of the complement, which holds in complex dimension $>2$ by Serre duality,

$$H_{\mathrm{cpt}}^2(X,\mathcal{O})\simeq H^{n-2}(X,K_X),$$

combined with Cartan’s Theorem B. When $n=2$, this dual group becomes $H^0(X,K_X)$, which need not vanish. In Hodge-theoretic terms, the cohomological obstruction that occurs in degree $n-2$ for $n>2$ collapses to degree 0 in the surface case, so the positivity and vanishing arguments effective in higher dimensions no longer produce a contradiction in ${\mathbb{CP}}^2$.

As a result, the existence of closed Levi-flat hypersurfaces in ${\mathbb{CP}}^2$ remains an open problem, despite strong analogies with higher dimensions suggesting that no such objects should exist. Shafikov’s survey also compiles significant partial results: topological constraints on hypothetical Levi-flats, nonexistence under global pluriharmonic defining functions, and restrictions arising from curvature and holonomy. It further situates the problem within broader contexts such as minimal sets of holomorphic foliations and connections to Stein fillability in contact geometry.

Recent Progress (2022–2025)

Beyond the classical results for $n>2$, several developments have sharpened the projective picture in recent years. First, a projective Chow-type theorem shows that (under mild singularity and algebraicity assumptions on the leaves) a real-analytic Levi-flat hypersurface in ${\mathbb{CP}}^n$ is tangent to the level sets of a rational function and hence is semialgebraic [4]. Second, curvature and foliation-based constraints continue to tighten: recent work relates Levi-flat geometry on complex space forms to quantitative curvature bounds on the hypersurface, complementing earlier Ricci curvature restrictions in ${\mathbb{CP}}^2$ [5,6]. Third, structural results for holomorphic foliations on projective manifolds (e.g., degree-four foliations on ${\mathbb{CP}}^n$) highlight transversely affine/projective alternatives that interface directly with foliation-based nonexistence strategies [7]. We also note the updated 2025 survey synthesizing these directions and clarifying why all three standard proofs fail specifically in $n=2$ [1].

Motivated by this rich backdrop, the goal of this paper is to examine the problem under an additional natural assumption: symmetry. Specifically, we consider Levi-flat hypersurfaces in ${\mathbb{CP}}^2$ that are homogeneous under a transitive effect by holomorphic automorphisms. By leveraging the explicit classification of homogeneous CR-manifolds from our prior work [8], we analyze the Levi foliations and the structure of such hypersurfaces in projective spaces. We then prove that no homogeneous Levi-flat hypersurface exists in ${\mathbb{CP}}^2$. This result extends the realm of known nonexistence theorems into settings with intrinsic geometric symmetry, thereby partially resolving the classical question under natural and meaningful constraints.

2. Homogeneous CR-Manifolds

A CR-manifold of type $(n,k)$ is a smooth real manifold M of dimension $m=2n+k$, equipped with a complex subbundle $\mathcal{H}\subset T_{\mathbb{C}}M:=TM{\otimes }_{\mathbb{R}}\mathbb{C}$ of complex rank n, called the CR-structure, which satisfies $\mathcal{H}\cap \overline{\mathcal{H}}=\left\{0\right\}$ and is involutive in the sense that $[X,Y]\in \Gamma \left(\mathcal{H}\right)$ for all local sections $X,Y\in \Gamma \left(\mathcal{H}\right)$. Here, $k=m-2n$ is called the CR-codimension. In the special case $n=0$ (so $k=m$), the structure $\mathcal{H}$ is referred to as totally real. It is worth noting that not every abstract CR-manifold can be realized as a CR submanifold of a complex manifold. When such a complex manifold X exists, meaning M embeds as a generic CR submanifold of X, we call X a complexification of M. (For examples and obstructions to such embeddings, see, e.g., [9,10].) A smooth map $f:(M,{\mathcal{H}}_M)\to (N,{\mathcal{H}}_N)$ between CR-manifolds is called a CR map if it preserves the CR structures, meaning $f_{*}\left({\mathcal{H}}_M\right)\subseteq {\mathcal{H}}_N$. In other words, at each point $p\in M$, the differential maps ${\mathcal{H}}_M{|}_p$ into ${\mathcal{H}}_N{|}_{f\left(p\right)}$. When such a map is a submersion, it is referred to as a CR submersion. A CR fiber bundle is a CR submersion

$$\pi :(E,{\mathcal{H}}_E)\to (B,{\mathcal{H}}_B)$$

whose fibers are CR-submanifolds of E and are all CR-diffeomorphic to a fixed CR-manifold F; in particular, the CR structure on E restricts to the given CR structure on each fiber. CR fiber bundles will be used to decompose a given CR-manifold into lower CR-codimension manifolds and, in particular, to describe fibrations of foliated CR-manifolds by their complex leaves. This perspective is especially useful in understanding the interplay between the geometry of the total space, the base, and the fibers. The following codimension lemma, established in [8], gives precise relations between the dimensions and CR-codimensions of the total space, base, and fibers in a CR fiber bundle. This is particularly useful when analyzing nontrivial fibrations where the total space E is a hypersurface (CR-codimension one), which forces either the base or the fibers to be complex manifolds (i.e., of CR-codimension zero).

Lemma 1 

(Codimension Lemma [8]). Let $(E,{\mathcal{H}}_E)$ and $(B,{\mathcal{H}}_B)$ be CR-manifolds, and suppose there exists a CR submersion

$$\pi :E⟶B.$$

Then each fiber F inherits a natural CR-structure given by

$${\mathcal{H}}_F\cong ker{\pi }_{*}{|}_{{\mathcal{H}}_E}.$$

Moreover,

• ${dim}_{\mathbb{R}}E={dim}_{\mathbb{R}}F+{dim}_{\mathbb{R}}B$,
• $\mathrm{codim}E=\mathrm{codim}F+\mathrm{codim}B$.

Given a CR-manifold $(M,\mathcal{H})$, the distribution $\mathcal{H}$ refers to the smooth assignment $p\mapsto {\mathcal{H}}_p\subset T_p^{\mathbb{C}}M$ of a complex subspace of the complexified tangent space at each point $p\in M$. The distribution $\mathcal{H}$ is not necessarily involutive, meaning that the Lie bracket of any two local sections of $\mathcal{H}$ does not necessarily remain in $\mathcal{H}$. To measure this failure, we introduce the Levi form, defined as the vector-valued 2-form.

$$L:\mathcal{H}\times \mathcal{H}\to T^{\mathbb{C}}M/(\mathcal{H}+\overline{\mathcal{H}}),L({\zeta }_x,{\xi }_x):={\pi }_x\left({[\zeta ,\overline{\xi }]}_x\right),$$

where $\pi$ is the natural projection of $T^{\mathbb{C}}M$ onto the quotient bundle $T^{\mathbb{C}}M/(\mathcal{H}+\overline{\mathcal{H}})$. Here, by a quotient bundle of a vector bundle E by a subbundle F, we mean the bundle whose fiber at each point $p\in M$ is the quotient vector space $E_p/F_p$, equipped with the natural projection $E\to E/F$.

If L vanishes identically, then the extended distribution $\mathcal{K}:=\mathcal{H}+\overline{\mathcal{H}}$ is involutive. By the Frobenius theorem (Chapter 1 of [11]), involutivity of $\mathcal{K}$ is equivalent to the existence of a unique maximal complex submanifold through each point whose tangent space at every point coincides with $\mathcal{K}$. In this setting, $(M,\mathcal{H})$ is called Levi-flat, and the Frobenius theorem ensures that M is locally foliated by complex submanifolds—the Levi leaves—tangent to $\mathcal{H}$; see also [12] for background on foliations in this context.

For the important case of CR-hypersurfaces (i.e., when $k=1$), and when M is locally given as

$$M=\{z\in X:\rho (z)=0\}$$

for a smooth real function $\rho$ with $d\rho \ne 0$ on M, the Levi form becomes a Hermitian form on $T_p^{1,0}M:={\mathcal{H}}_p$ at each $p\in M$ given explicitly by

$${\mathcal{L}}_p(V,\overline{W})=-i\partial \overline{\partial }\rho (V,\overline{W})=-i\sum _{j,k}{\displaystyle \frac{{\partial }^2\rho }{\partial z_j\partial {\overline{z}}_k}}{V}^j\overline{W^k}.$$

The condition ${\mathcal{L}}_p\equiv 0$ then precisely characterizes Levi-flatness, ensuring that $\mathcal{H}$ integrates locally to complex leaves. For further details on the general theory of CR-manifolds, Levi forms, their local and global properties, and homogeneous structures, we refer the reader to [9,10,13].

• Real-analytic Levi-flat CR-manifolds: local description

A CR-manifold $(M,\mathcal{H})$ is called real-analytic if both the manifold M and the CR distribution $\mathcal{H}$ are real-analytic. A fundamental theorem of Andreotti and Fredricks [13] ensures that every real-analytic CR-manifold admits a complexification. In the special case of real-analytic Levi-flat CR-manifolds, there is a particularly transparent local structure. Freeman’s theorem [14] states that any such manifold of type $(n,k)$ is locally biholomorphic to ${\mathbb{R}}^k\times {\mathbb{C}}^n$, with the Levi foliation given by the complex slices $\left\{x\right\}\times {\mathbb{C}}^n$ for $x\in {\mathbb{R}}^k$. This makes both the CR submanifold structure inside its complexification and the foliation by complex manifolds entirely explicit.

A homogeneous CR-manifold is a CR-manifold $(M,\mathcal{H})$ on which a Lie group G acts transitively by CR-automorphisms, meaning diffeomorphisms preserving the CR structure $\mathcal{H}$. Such manifolds are automatically real analytic since $\mathcal{H}$ is locally spanned by real-analytic vector fields arising from the action of the Lie algebra $\mathfrak{g}$ of G. This ensures that every homogeneous CR-manifold admits a local complexification.

Each element of the complexified Lie algebra $\widehat{\mathfrak{g}}=\mathfrak{g}\oplus i\mathfrak{g}$ acts locally on a complexification X as a holomorphic vector field, so the associated connected complex Lie group $\widehat{G}$ acts locally holomorphically on X. The main interest is in cases where this local action extends to a global holomorphic action of $\widehat{G}$ that is also transitive on X so that M appears as a G-orbit in a single complex homogeneous space $X=\widehat{G}/\widehat{H}$ with $\widehat{H}\cap G=H$. In this case, X is called a globalization of M. Transitivity of $\widehat{G}$ on X is essential: without it, the complexification may decompose into several complex orbits, and M may fail to embed as a real form of a single complex homogeneous space.

Not every homogeneous CR-manifold is globalizable, and the following example illustrates this failure. Consider the 2-dimensional affine quadric

$$X:=\widehat{G}/\widehat{J},$$

where $\widehat{G}={\mathrm{SL}}_2\left(\mathbb{C}\right)$ and $\widehat{J}$ is the subgroup of diagonal matrices. The subgroup $\widehat{J}$ contains $\{\pm \mathrm{Id}\}$ so that $\widehat{G}$ acts with this small ineffectivity. Let $x_0\in X$ be a neutral point where $\widehat{J}={\widehat{G}}_{x_0}$. The unipotent group $\widehat{U}\cong \mathbb{C}$ of upper-triangular matrices realizes X as the total space of the principal $\mathbb{C}$-bundle $\widehat{G}/\widehat{J}\to \widehat{G}/\widehat{J}\widehat{U}\cong {\mathbb{P}}^1\left(\mathbb{C}\right)$. Identifying the $\widehat{U}$-orbit $\widehat{U}·x_0$ with $\mathbb{C}$, $\Sigma$ is defined as the subset of X corresponding to ${\mathbb{R}}^{\ge 0}$. Now take $G={\mathrm{SU}}_2$ and note that every G-orbit in X intersects $\Sigma$ in exactly one point. The G-orbit $M_{x_0}$ of the neutral point is a copy of the 2-sphere, embedded as a totally real submanifold. For $x\in \Sigma \setminus \left\{x_0\right\}$, the orbit $M_x=G·x$ is a hypersurface, and since $G_x$ is just the ineffectivity $\{\pm \mathrm{Id}\}$, $M_x$ is simply the group ${\mathrm{PSU}}_2$ equipped with a left-invariant CR structure. Consider the universal cover $\tilde{Z}$ of the complement Z of $M_{x_0}$ in X. Here G acts freely as a group of holomorphic transformations. The slice $\Sigma$ lifts to a slice $\tilde{\Sigma }$ for the G-action, and for $\tilde{x}\in \tilde{\Sigma }$, the CR-homogeneous space ${\tilde{M}}_x$ is just G equipped with a left-invariant CR structure. This ${\tilde{M}}_x$ is an example of a strongly pseudoconvex hypersurface which cannot be filled in to a Stein space and thus cannot be globalized in our sense: if there were a globalization $\widehat{G}/\widehat{H}$, then $\widehat{H}$ would have to be a subgroup of $\widehat{J}$, forcing $\widehat{H}=\widehat{J}$ and hence $M_x=M_{x_0}$, a contradiction. This demonstrates concretely how failure of the transitivity condition for $\widehat{G}$ obstructs globalization.

For further details on these constructions and their implications for Levi-flat geometry, see [8,15,16,17,18].

One of the powerful tools in the study of complex homogeneous manifolds is the so-called normalizer fibration, originally developed in the context of complex transformation groups (see [17,19]). This result can also be applied in the setting of homogeneous CR-manifolds by considering restrictions to a real subgroup $G\subset \widehat{G}$.

Theorem 1 

(Normalizer fibration). Let $\widehat{G}$ be a connected complex Lie group and $\widehat{H}$ a closed complex Lie subgroup with identity component $\widehat{H}°$. Then the normalizer

$$\widehat{N}:=N_{\widehat{G}}\left(\widehat{H}°\right)$$

induces a holomorphic fibration

$$\widehat{G}/\widehat{H}⟶\widehat{G}/\widehat{N},$$

whose fibers $\widehat{N}/\widehat{H}$ are parallelizable and whose base $\widehat{G}/\widehat{N}$ is an orbit of $\mathrm{Ad}\left(\widehat{G}\right)$ in a projective variety. Here, $\mathrm{Ad}\left(\widehat{G}\right)$ denotes the adjoint representation of $\widehat{G}$ on its Lie algebra $\widehat{\mathfrak{g}}$, given by $\mathrm{Ad}\left(g\right)\left(X\right)=gX{g}^{-1}$ for $g\in \widehat{G}$ and $X\in \widehat{\mathfrak{g}}$. A projective variety is a closed complex-analytic subvariety of complex projective space ${\mathbb{CP}}^n$, defined as the common zero locus of a collection of homogeneous polynomials.

2.1. Homogeneous Structure of the Levi Foliation

Let $M=G/H$ be a homogeneous Levi-flat CR-manifold with Levi foliation $\mathcal{F}$. In this subsection, we will show that the leaves of such homogeneous Levi-flat CR-manifolds are biholomorphic to each other, and moreover, they arise as orbits of certain Lie subgroups of G. For further details on these results and their proofs in the context of homogeneous Levi-flat CR-geometry, we refer the reader to [8].

Lemma 2. 

Let Λ be any leaf of the Levi foliation $\mathcal{F}$. Then for any $g\in G$, either

$$g·\Lambda =\Lambda \mathrm{or}g·\Lambda \cap \Lambda =\varnothing .$$

Proof. 

Assume $g·\Lambda \cap \Lambda \ne \varnothing$, and set

$$Y:=\{y\in \Lambda :g·y\in \Lambda \}.$$

We want to show that $Y=\Lambda$ by proving it is both closed and open in $\Lambda$.

Closedness. Take a sequence $\left\{y_n\right\}\subset Y$ with $y_n\to y$ in $\Lambda$. We wish to prove $g·y\in \Lambda$, i.e., $y\in Y$. Choose a local leaf chart W around y, with $\phi (\Lambda \cap W)=\phi \left(W\right)\cap {\mathbb{R}}^d,$ where $\phi :W\to {\mathbb{R}}^n$ is a local coordinate map so that

$$\Lambda \cap W=F^{-1}\left(0\right),\mathrm{with}F:=\pi \circ \phi \mathrm{and}\pi :{\mathbb{R}}^n\to {\mathbb{R}}^{n-d}\mathrm{the}\mathrm{standard}\mathrm{projection}.$$

Thus $\Lambda \cap W$ is a closed subset of $\Lambda$, which implies that $\Lambda \cap gW$ is also closed in $\Lambda$. Therefore, since the sequence $\{g·y_n\}$ lies in $\Lambda \cap gW$ for sufficiently large n and converges to $g·y$, we have $g·y\in \Lambda$. This shows that $y\in Y$, and hence Y is closed.

Openness. Let $x\in Y$, and take a small neighborhood U of x in $\Lambda$. Since g is a CR-automorphism, it maps U to a local open complex manifold $g\left(U\right)$ in M. By Freeman’s theorem [14], $\Lambda$ is locally the unique maximal complex submanifold through x. Thus, since $g\left(U\right)\cap \Lambda \ne \varnothing$, we must have $g\left(U\right)\subset \Lambda$. This means $U\subset Y$, so Y is open. Moreover, by connectedness of $\Lambda$, we conclude $Y=\Lambda$; hence $g·\Lambda =\Lambda$. □

Corollary 1 

(Leaf stabilizer). Let Λ be the leaf in the Levi foliation $\mathcal{F}$ of M through the base point $p_0=eH\in M$. Then there exists a connected (not necessarily closed) Lie subgroup K of G, called the leaf stabilizer, such that

$$\Lambda =K·p_0.$$

Moreover, for each $g\in G$, the group $gK{g}^{-1}$ stabilizes the leaf $g·\Lambda$ through the point $p_1:=g·p_0$ so that

$$\mathcal{F}={\left\{g·\Lambda \right\}}_{g\in G}={\left\{gK{g}^{-1}·p_0\right\}}_{g\in G}.$$

We also observe that the restriction of the G-action to each leaf $\Lambda$ is holomorphic. In particular, this implies that all leaves of the Levi foliation are biholomorphic to each other, which is a remarkably strong conclusion. Thus, if one leaf is compact, then by homogeneity all leaves are compact; if one leaf is dense in M, then all leaves are dense.

Corollary 2. 

Let Λ be the leaf of the Levi foliation $\mathcal{F}$ through the base point $p_0\in M=G/H$, and let K be the leaf stabilizer defined previously. Then the isotropy subgroup H stabilizes Λ. Consequently, the connected component $H^0$ of H is contained in K, and moreover H normalizes K. In particular, $KH$ is a Lie subgroup of G.

Proof. 

Since $p_0\in \Lambda$, we have $p_0\in \Lambda \cap h·\Lambda$ for all $h\in H$. By the leaf-separation Lemma 2, this forces $h·\Lambda =\Lambda$ for all $h\in H$. Therefore the connected component $H^0$ lies inside K.

Moreover, for any $h\in H$, the conjugate subgroup $hK{h}^{-1}$ stabilizes the leaf

$$hK{h}^{-1}·(h·p_0)=h·(K·p_0)=h·\Lambda =\Lambda .$$

By the uniqueness of the leaf stabilizer subgroup, we conclude $hK{h}^{-1}=K$. Thus H normalizes K, and so $KH$ is a Lie subgroup of G. □

Corollary 3. 

Let $G_c$ be the connected complex Lie subgroup of G with Lie algebra ${\mathfrak{g}}_c:=\mathfrak{g}\cap i\mathfrak{g}.$ Then $G_c\subset K$.

Proof. 

The orbit $G_c·p_0$ is a complex submanifold of M. By Freeman’s local theorem on Levi-flat CR-manifolds, the leaf $\Lambda$ is the unique maximal complex submanifold through $p_0$. Thus locally $G_c·p_0\subset \Lambda$. By repeating the local process along the $G_c$ orbit, we have the global inclusion. □

Corollary 4 

(Leaf reduction). If Λ is compact (and hence all leaves of $\mathcal{F}$ are compact), then the Lie subgroup $KH$ is closed in G. This gives rise to a fiber bundle, often called the leaf reduction bundle,

$$M=G/H⟶G/KH=:M/\mathcal{F},$$

whose fibers are exactly the leaves of the compact complex Levi foliation $\mathcal{F}$. Moreover, the base $G/KH=M/\mathcal{F}$ is a totally real homogeneous CR-manifold of dimension equal to the codimension of M and is referred to as the leaf space.

We now carry out a similar analysis at the level of the complex Lie group $\widehat{G}$. Suppose $M=G/H$ admits a globalization $X:=\widehat{G}/\widehat{H}$, making X a complex homogeneous manifold containing M as a real submanifold. As defined above, let K denote the leaf stabilizer group in G for the Levi foliation, with Lie algebra $\mathfrak{k}$.

Lemma 3. 

Let $\widehat{K}$ be the connected complex Lie subgroup of $\widehat{G}$ corresponding to the complex Lie subalgebra $\widehat{\mathfrak{k}}:=\mathfrak{k}+i\mathfrak{k}$ of $\widehat{\mathfrak{g}}$. Then

• The leaf Λ through the base point $p_0$ is precisely the holomorphic orbit of $\widehat{K}$ in X.
• The connected component ${\widehat{H}}^0$ of the complex isotropy subgroup $\widehat{H}$ is contained in $\widehat{K}$.

Proof. 

• This follows directly from the definition of $\widehat{K}$.
• At the Lie algebra level, by Corollary 3, we have

  $${\mathfrak{g}}_c:=\mathfrak{g}\cap i\mathfrak{g}\subset \mathfrak{k}.$$
  Thus

  $$\begin{array}{cc}\hfill \widehat{\mathfrak{g}}& =\left(\mathfrak{g}/{\mathfrak{g}}_c\right)\oplus i\left(\mathfrak{g}/{\mathfrak{g}}_c\right)\oplus {\mathfrak{g}}_c,\hfill \\ \hfill \widehat{\mathfrak{k}}& =\left(\mathfrak{k}/{\mathfrak{g}}_c\right)\oplus i\left(\mathfrak{k}/{\mathfrak{g}}_c\right)\oplus {\mathfrak{g}}_c.\hfill \end{array}$$
  Hence the complex codimension of $\widehat{\mathfrak{k}}$ in $\widehat{\mathfrak{g}}$ equals the real codimension of $\mathfrak{k}$ in $\mathfrak{g}$,

  $$\begin{array}{cc}\hfill {dim}_{\mathbb{C}}\widehat{G}-{dim}_{\mathbb{C}}\widehat{K}& ={dim}_{\mathbb{R}}G-{dim}_{\mathbb{R}}K\hfill \\ \hfill & \stackrel{\mathrm{Corollary}}{=}{dim}_{\mathbb{R}}{G}_c-{dim}_{\mathbb{R}}\Lambda \hfill \\ \hfill & ={dim}_{\mathbb{C}}X-{dim}_{\mathbb{C}}\Lambda \hfill \\ \hfill & ={dim}_{\mathbb{C}}(\widehat{G}/\widehat{H})-{dim}_{\mathbb{C}}\left(\widehat{K}/(\widehat{K}\cap \widehat{H})\right).\hfill \end{array}$$
  Therefore,

  $${dim}_{\mathbb{C}}\widehat{H}={dim}_{\mathbb{C}}\left(\widehat{K}\cap \widehat{H}\right),$$

  which implies ${\widehat{H}}^0\subset \widehat{K}$, as required.

□

Remark 1. 

One may also define the complex stabilizer of the leaf Λ in the previous lemma as the (possibly disconnected) complex Lie group

$$\widehat{J}:=\left\{g\in \widehat{G}:g(\Lambda )\subset \Lambda \right\}.$$

Clearly, this satisfies $\widehat{H}°\subset \widehat{J}$ and $H\subset \widehat{J}$. For simplicity of notation, we will henceforth abuse notation slightly and write $\widehat{K}$ in place of $\widehat{J}$.

2.2. Minimality Condition

The globalization $\widehat{G}/\widehat{H}$ is generally not unique. For instance, if $G/H=S^1$, then $\widehat{G}/\widehat{H}={\mathbb{C}}^{*}$, $\widehat{G}/\widehat{H}=T$ (a one-dimensional complex torus), and $\widehat{G}/\widehat{H}={\mathbb{CP}}^1$ are all valid globalizations. Similar phenomena can arise in more intricate examples. To address this ambiguity, we impose the minimality condition, requiring that $\widehat{H}=H\widehat{H}°,$ where $\widehat{H}°$ is the identity component of $\widehat{H}$. This can always be arranged since the quotient $\widehat{H}/\left(H\widehat{H}°\right)$ is discrete and the natural map

$$\widehat{G}/\left(H\widehat{H}°\right)⟶\widehat{G}/\widehat{H}$$

is a covering that is biholomorphic over M. We then replace $\widehat{H}$ by $H\widehat{H}°$ to enforce minimality.

Recall from Lemma 3 that $\widehat{H}°\subset \widehat{K}$, where $\widehat{K}$ denotes the (possibly disconnected) complex stabilizer of the leaf $\Lambda$ in $\widehat{G}$ (cf. Remark 1). Thus, under the minimality assumption it follows that $\widehat{H}\subset \widehat{K}$. In situations where the closure properties of $\Lambda$ require careful control, we may pass to an appropriate covering and work with $\widehat{H}\widehat{K}°=H\widehat{K}°$, which serves as a well-defined stabilizer subgroup incorporating both $\widehat{H}$ and the connected component $\widehat{K}°$ of the stabilizer.

3. Projective Homogeneous CR-Manifolds

For the purposes of this paper, we now focus on the projective setting, studying Levi-flat hypersurfaces that appear as homogeneous CR-manifolds embedded in complex projective space ${\mathbb{CP}}^n$.

Let G be a connected real Lie group with a faithful representation into the automorphism group of ${\mathbb{CP}}^n$, that is, $G\subset {\mathrm{PSL}}_{n+1}\left(\mathbb{C}\right)$ as a Lie subgroup (not necessarily closed). Suppose M is a compact Levi-flat homogeneous CR-manifold on which G acts transitively and almost effectively by CR-automorphisms. Assume further that M is G-equivariantly embedded in ${\mathbb{CP}}^n$. Then we can write

$$M=G·p_0\hookrightarrow {\mathbb{CP}}^n\mathrm{for}\mathrm{some}\mathrm{point}{p}_0\in {\mathbb{CP}}^n$$

Let $\widehat{G}$ denote the smallest connected complex Lie subgroup of ${\mathrm{PSL}}_{n+1}\left(\mathbb{C}\right)$ that contains G. In other words, $\widehat{G}$ is the connected complex Lie subgroup of ${\mathrm{PSL}}_{n+1}\left(\mathbb{C}\right)$ corresponding to the complexified Lie algebra $\widehat{\mathfrak{g}}:=\mathfrak{g}+i\mathfrak{g}$, where $\mathfrak{g}$ is the Lie algebra of G.

Let $\widehat{H}$ be the complex isotropy subgroup of $\widehat{G}$ at the point $p_0$. Then the orbit

$$X:=\widehat{G}·p_0=\widehat{G}/\widehat{H}$$

is a complex homogeneous manifold containing M as a real submanifold. This yields the sequence of equivariant embeddings

$$M=G/H\hookrightarrow X=\widehat{G}/\widehat{H}\hookrightarrow {\mathbb{CP}}^n.$$

Consider a Levi decomposition of $\widehat{G}$,

$$\widehat{G}=\widehat{S}⋉\widehat{R},$$

where $\widehat{S}$ is a maximal connected semisimple subgroup and $\widehat{R}$ is the radical. We denote by $\widehat{\mathfrak{s}}$ and $\widehat{\mathfrak{r}}$ their respective Lie algebras.

The following theorem, proved in [8], will be fundamental for our discussion.

Theorem 2 

([8]). The radical $\widehat{R}$ is contained in the center of $\widehat{G}$ so that

$$\widehat{G}=\widehat{S}\times \widehat{R}$$

(possibly up to finite intersection). Moreover, the real group G splits accordingly as

$$G=S\times R,$$

where S and R are the real forms corresponding to $\widehat{S}$ and $\widehat{R}$, respectively.

3.1. Leaves Are Flag Manifolds

Our aim is to show that in the projective setting, each leaf $\Lambda$ of the Levi foliation $\mathcal{F}$ on M is a homogeneous rational manifold, more precisely, a flag manifold.

In fact, it suffices to establish that $\Lambda$ is compact. By the Borel–Remmert structure theorem, any compact connected complex homogeneous Kähler manifold decomposes as a product of a compact complex torus and a flag manifold (see [20]). However, by the Borel fixed point theorem, no positive-dimensional compact complex torus can embed equivariantly in projective space ${\mathbb{CP}}^n$ (see, e.g., [21]). It follows that $\Lambda$ must itself be a flag manifold.

Definition 1. 

A flag manifold is a homogeneous complex manifold of the form $S/P$, where S is a connected complex semisimple Lie group and $P\subset S$ is a parabolic subgroup. Equivalently, flag manifolds are the closed S-orbits in projective algebraic S-varieties.

Theorem 3. 

Each leaf Λ of the Levi foliation $\mathcal{F}$ is a flag manifold.

Proof. 

Let $X=\widehat{G}/\widehat{H}$ be the ambient complex homogeneous space containing the homogeneous Levi-flat CR-hypersurface M, and let $\Lambda \subset M$ be a (nontrivial) leaf of the Levi foliation. Write the Levi decomposition $\widehat{G}=\widehat{S}⋉\widehat{R}$, where $\widehat{S}$ is semisimple and $\widehat{R}$ is the solvable radical. For a point $p\in \Lambda$, let ${\widehat{G}}_{\Lambda }^{\circ }$ be the connected complex subgroup of $\widehat{G}$ generated by holomorphic vector fields tangent to $\Lambda$; then $\Lambda ={\widehat{G}}_{\Lambda }^{\circ }·p$ as a complex homogeneous space.

Induction on the codimension $k:={\mathrm{codim}}_M(\Lambda )$. If $k=0$, then $\Lambda =M$ is a complex homogeneous hypersurface in X. Since $\widehat{S}$ acts algebraically on a projective completion of X, the closed $\widehat{S}$-orbit through p is projective and hence a flag manifold, and $\Lambda$ is (biholomorphic to) that orbit. This settles the base case. Assume the theorem holds for all leaves of codimension $<k$ and let ${\mathrm{codim}}_M(\Lambda )=k\ge 1$. We consider three cases.

Case 1: $\widehat{S}$ does not act transitively on X. Since $\widehat{S}$ is reductive and acts algebraically, there is the commutator (or Abelian) fibration

$$\pi :X=\widehat{G}/\widehat{H}⟶\widehat{G}/\left({\widehat{G}}^{\prime }\widehat{H}\right),$$

whose base is a Stein Abelian group. The $\widehat{R}$-orbits are the fibers of $\pi$. The image $\pi \left(M\right)$ is a totally real R-orbit (a product of circles), so the neutral fiber $F:={\pi }^{-1}\left(\pi \left(p\right)\right)$ is $\widehat{S}$-invariant and contains $\Lambda$. Because $\pi$ is a CR-submersion along M, the codimension of $\Lambda$ in $F\cap M$ is strictly smaller than k. By the induction hypothesis applied to the homogeneous CR-manifold $F\cap M\subset F$, the leaf $\Lambda$ (viewed inside F) is a flag manifold. Hence $\Lambda$ is a flag manifold in M as well.

Case 2: $\widehat{S}$ acts transitively on X and $\widehat{R}\ne \left\{e\right\}$. Let $\widehat{Z}:=Z_{\widehat{L}}\left(\widehat{S}\right)$ be the centralizer of $\widehat{S}$ in an algebraic stabilizer $\widehat{L}$ of a projective completion of X. Then $\widehat{Z}$ is a complex algebraic Abelian group acting algebraically on X. Consider the quotient map

$$\pi :X⟶X/\widehat{Z}.$$

Each $\pi$-fiber is an orbit of $\widehat{Z}$ and is biholomorphic to ${\mathbb{C}}^s\times {\left({\mathbb{C}}^{*}\right)}^t$ for some $s,t$. Since $\Lambda$ lies in a compact subset of M, its intersection with any $\pi$-fiber is discrete (a positive-dimensional analytic subset in ${\mathbb{C}}^s\times {\left({\mathbb{C}}^{*}\right)}^t$ would be noncompact). Thus $Q:=\pi (\Lambda )$ is a complex submanifold of $\pi \left(M\right)$ with

$${\mathrm{codim}}_{\pi \left(M\right)}\left(Q\right)<{\mathrm{codim}}_M(\Lambda )=k.$$

By the induction hypothesis applied in $X/\widehat{Z}$, the leaf Q is a flag manifold (indeed, a homogeneous rational variety). The ${\widehat{G}}_{\Lambda }$ equivariance of $\pi$ gives a holomorphic covering $\Lambda \to Q$. Since flag manifolds are simply connected, this covering is injective and hence a biholomorphism; therefore $\Lambda \simeq Q$ is a flag manifold.

Case 3: $\widehat{G}=\widehat{S}$ (semisimple, $\widehat{R}=\left\{e\right\}$). As $\widehat{S}$ acts algebraically, X admits a projective completion with finite fundamental group; passing to a finite cover (replacing $\widehat{H}$ by $\widehat{H}\widehat{H}°$) does not change the argument. Let

$$\widehat{N}:=N_{\widehat{G}}\left({\widehat{G}}_{\Lambda }^{\circ }\right)$$

be the normalizer of ${\widehat{G}}_{\Lambda }^{\circ }$ in $\widehat{G}$, and consider the (normalizer) fibration $X\to X/\widehat{N}$.

(3a) If ${\widehat{G}}_{\Lambda }^{\circ }$ is not normal in $\widehat{S}$. Then the neutral fiber of $X\to X/\widehat{N}$ is a proper $\widehat{S}$-invariant complex submanifold of X containing $\Lambda$. As in Case 1, the codimension of $\Lambda$ inside that fiber is strictly smaller than k, so the induction hypothesis applies, and $\Lambda$ is a flag manifold.

(3b) If ${\widehat{G}}_{\Lambda }^{\circ }$ is normal in $\widehat{S}$. Then ${\widehat{G}}_{\Lambda }^{°}$ is semisimple and acts algebraically. Its orbits are Zariski-open in their closures; hence $\Lambda$ is a closed $\widehat{S}$-orbit in a projective $\widehat{S}$-variety. Closed $\widehat{S}$-orbits in projective algebraic $\widehat{S}$-varieties are flag manifolds (stabilizers are parabolic). Thus $\Lambda$ is a flag manifold.

In all cases, $\Lambda$ is a flag manifold. This completes the induction and the proof. □

Remark 2. 

At this stage, we can conclude that there do not exist homogeneous Levi-flat hypersurfaces in ${\mathbb{CP}}^2$. Indeed, by Theorem 3 every leaf of the Levi foliation is a flag manifold and hence a compact complex homogeneous submanifold. By Chow’s theorem [22] each such leaf is algebraic; in ${\mathbb{CP}}^2$ it is therefore a projective curve. By Bézout’s theorem, any two distinct projective plane curves intersect (counting multiplicities); see, e.g., ([23], Chap. I) or ([24], Chap. 1). This contradicts the disjointness of distinct leaves of a foliation. Consequently, no homogeneous Levi-flat hypersurface exists in ${\mathbb{CP}}^2$.

Nevertheless, we now go beyond mere existence questions and undertake a deeper structural analysis. Our goal is to describe what projective Levi-flat homogeneous hypersurfaces in ${\mathbb{CP}}^n$ actually look like—mapping out the allowable group actions, leaf geometries, and bundle structures—to provide a clearer and more complete picture.

3.2. Reduction to Semisimple Groups

In this section, for the purposes of this paper, we focus on the case most relevant to our analysis: CR submanifolds of complex projective space ${\mathbb{CP}}^n$. Since ${\mathbb{CP}}^n$ is compact, we restrict our study to the situation where $X=\widehat{G}/\widehat{H}$ is a compact complex homogeneous space. By the Borel–Remmert structure theorem, any compact homogeneous Kähler manifold is biholomorphic to a product $T\times Q$, where T is a complex torus and Q is a flag manifold. We recall the commutator fibration $\widehat{G}/\widehat{H}⟶\widehat{G}/{\widehat{G}}^{\prime }\widehat{H}$, whose base is a Stein manifold (cf [25]) on which the radical $\widehat{R}$ acts transitively. Because $\widehat{S}\cap \widehat{R}$ is finite, the $\widehat{R}$-orbits in the base are finite covers of Stein manifolds, and hence themselves Stein. This implies that the torus factor T cannot arise from the radical action. Thus we may assume that the radical acts trivially on X. In other words, we reduce to the case where $\widehat{G}=\widehat{S}$ is semisimple and acts transitively on X.

As before, we impose the minimality condition by replacing $\widehat{H}$ with $H\widehat{H}°$ (see Section 2.2), which only amounts to passing to a finite cover so that $\widehat{S}$ continues to act algebraically.

We also consider the leaf reductions given by Corollary 4:

$$\widehat{S}/\widehat{H}\stackrel{\Lambda }{\to }\widehat{S}/\widehat{K}=:Z,\mathrm{and}\mathrm{the}\mathrm{induced}\mathrm{reduction}\mathrm{for}M:S/H\stackrel{\Lambda }{\to }S/J=:Y,$$

where Z is the complex leaf space and Y is the real leaf space, which is a totally real homogeneous CR-submanifold of Z. Thus,

$${dim}_{\mathbb{R}}Y={dim}_{\mathbb{C}}Z=\mathrm{codim}M.$$

Recall also from Corollary 3 that the maximal connected complex subgroup M of S is contained in J. Thus we have the following lemma.

Lemma 4. 

The real leaf space Y is an orbit of a real form of the complex semisimple group $\widehat{S}$.

We also have the following important lemma.

Lemma 5. 

If Z is compact, then the real form S of $\widehat{S}$ cannot be compact.

Proof. 

If Z is compact, then it is simply connected. By Montgomery’s theorem [26], the compact real form of $\widehat{S}$ acts transitively on Z. This is impossible in our situation since we are assuming the codimension is positive, so the leaves cannot fill the entire space. □

In particular, since the only one-dimensional complex homogeneous manifold of a complex semisimple Lie group is ${\mathbb{CP}}^1$, we obtain the following classification in codimension one.

Theorem 4. 

If $M=S/H$ has codimension one in $X=\widehat{S}/\widehat{H}$, then

$$M=Q\times S^1\mathrm{and}X=Q\times {\mathbb{CP}}^1,$$

where $Q=\Lambda$ is a flag manifold.

Proof. 

The complex leaf space Z is one-dimensional; hence $Z={\mathbb{CP}}^1$ as a holomorphic ${\mathrm{SL}}_2\left(\mathbb{C}\right)$-orbit, and the real leaf space is $Y=S^1$, the corresponding ${\mathrm{SL}}_2\left(\mathbb{R}\right)$-orbit. The triviality of the leaf reduction follows from Lemma 5.6 of [8]. □

It follows that the only possible compact projective Levi-flat hypersurfaces arising in this setting are products of the form $Q\times S^1$ inside $Q\times {\mathbb{CP}}^1$, or, by the minimality condition, finite coverings thereof. These exhaust all possibilities for homogeneous Levi-flat hypersurfaces in this framework.

However, since ${\mathbb{CP}}^n$ is simply connected, it cannot be finitely covered by such a product. Thus, no Levi-flat homogeneous hypersurface of this type can exist inside ${\mathbb{CP}}^n$.

Theorem 5 

(Main Theorem). There do not exist homogeneous Levi-flat closed hypersurfaces in ${\mathbb{CP}}^n$ for any $n\ge 2$.