---
title: Jouanolou Foliation on Complex Projective Planes
url: https://www.emergentmind.com/topics/jouanolou-foliation
type: topic
---

# Jouanolou Foliation on Complex Projective Planes

Searching arXiv for recent papers on the Jouanolou foliation and related foliations.
The **Jouanolou foliation** generally refers to an explicit holomorphic foliation on the complex projective plane \(\mathbb{P}^2\) that became a canonical example of a degree-\(d\) foliation without algebraic invariant curves. In the modern literature, the name is used both for Jouanolou’s original characteristic-zero examples and for specific projective models such as
\[
\omega_d=(x^{d} z-y^{d+1})\,dx+(x y^{d}-z^{d+1})\,dy+(z^{d} y-x^{d+1})\,dz,
\]
or, on the affine chart \(z\neq 0\) with coordinates \((x,y)\),
\[
v_d=(x y^d-1)\partial_x-(x^d-y^{d+1})\partial_y,
\]
which define the same degree-\(d\) foliation on \(\mathbb{P}^2\) [2507.07277], [2305.06251]. Its significance is twofold. First, it supplies an explicit obstruction to algebraic integrability: for \(d>1\), it furnishes foliations with no algebraic invariant curve, a cornerstone of Jouanolou’s classical theory [2305.06251]. Second, it has become a focal example linking complex foliation theory, arithmetic reduction modulo \(p\), positive-characteristic \(p\)-divisors, holonomy, and dynamical invariants such as harmonic currents [2507.07277], [2503.09152].

## 1. Definition and projective realization

A holomorphic foliation by curves of degree \(d\) on \(\mathbb{P}^2=\mathbb{P}^2_{\mathbb{C}}\) can be represented by a homogeneous integrable \(1\)-form
\[
\omega\in H^0(\mathbb{P}^2,\Omega^1_{\mathbb{P}^2}(d+2))
\]
satisfying the Euler condition \(i_R\omega=0\) and the Frobenius integrability condition \(\omega\wedge d\omega=0\), where \(R=x\partial_x+y\partial_y+z\partial_z\) is the radial vector field [2507.07277]. In homogeneous coordinates \([x:y:z]\), one writes
\[
\omega=A\,dx+B\,dy+C\,dz,
\]
with \(A,B,C\) homogeneous of degree \(d+1\) and \(xA+yB+zC=0\) [2305.06251].

For the Jouanolou foliation of degree \(d\ge 2\), the affine vector field on the chart \(z\neq 0\) is
\[
v_d=(x y^d-1)\partial_x-(x^d-y^{d+1})\partial_y,
\]
with associated affine \(1\)-form
\[
\omega_{\mathrm{aff}}=(x^d-y^{d+1})\,dx+(x y^d-1)\,dy.
\]
A homogeneous extension to \(\mathbb{P}^2\) is
\[
\omega_{J,d}=(X^d Z-Y^{d+1})\,dX+(X Y^d-Z^{d+1})\,dY-(X^{d+1}-Y Z^d)\,dZ,
\]
which satisfies \(i_R\omega_{J,d}=0\) and restricts to \(\omega_{\mathrm{aff}}\) on \(Z=1\) [2507.07277]. Equivalent projective formulas also appear in the arithmetic literature under the notation
\[
\omega_d=(x^{d} z-y^{d+1})\,dx+(x y^{d}-z^{d+1})\,dy+(z^{d} y-x^{d+1})\,dz
\]
[2305.06251].

A closely related degree-\(2\) model is given by the homogeneous vector field
\[
J_2(x,y,z)=y^2\partial_x+z^2\partial_y+x^2\partial_z,
\]
whose projectivization defines the degree-\(2\) Jouanolou foliation; the corresponding \(1\)-form is
\[
\omega=(z^3-x^2y)\,dx+(x^3-y^2z)\,dy+(y^3-xz^2)\,dz
\]
[2303.09313]. This suggests that the literature uses several equivalent normal forms, depending on whether one emphasizes affine vector fields, homogeneous \(1\)-forms, or projectivized polynomial vector fields.

## 2. Algebraic invariant curves and non-algebraicity

An irreducible algebraic curve \(C\subset \mathbb{P}^2\), defined by a homogeneous polynomial \(F\), is invariant by a foliation \(\mathcal{F}\) defined by \(\omega\) if and only if
\[
\omega\wedge dF=F\eta
\]
for some homogeneous \(2\)-form \(\eta\); equivalently, \(F\) divides \(\omega\wedge dF\) in the homogeneous coordinate ring [2507.07277]. In affine language, this is the standard tangency criterion for an invariant algebraic leaf.

Jouanolou’s classical theorem asserts that over \(\mathbb{C}\), the Jouanolou foliation \(\mathcal{I}_d\) has no algebraic solutions for every \(d>1\) [2305.06251]. In the later arbitrary-field Darboux–Jouanolou framework, this example serves as the basic witness that algebraic non-integrability is genuinely possible and that degree-dependent thresholds for integrability are essential [1902.09571], [2102.00520].

The contrast with Darboux–Jouanolou integrability is precise. For a polynomial \(1\)-form \(\omega\), sufficiently many invariant irreducible hypersurfaces force a rational first integral. In characteristic \(0\), for \(r=1\) and \(n=2\), the threshold is \(d(d+1)/2+2\) invariant curves for a rational first integral [2102.00520]. The Jouanolou foliation lies at the opposite extreme: it has no algebraic invariant curve at all [2305.06251]. A plausible implication is that the Jouanolou example is not merely exceptional but structurally central in the geometry of the parameter space of foliations, because a single explicit example suffices to establish density phenomena for non-algebraicity.

## 3. Arithmetic reduction and the \(p\)-divisor

A major recent development is the arithmetic study of the Jouanolou foliation via reduction modulo \(p\). If \(\mathcal{F}\) is defined over a finitely generated \(\mathbb{Z}\)-algebra \( \mathbb{Z}[\mathcal{F}] \), one can reduce its defining coefficients modulo maximal ideals \(\mathfrak p\) and obtain a foliation \(\mathcal{F}_{\mathfrak p}\) over a finite field of characteristic \(p\) [2507.07277], [2305.06251]. This provides a bridge between complex foliations and positive-characteristic foliation theory.

In characteristic \(p>0\), one considers \(p\)-closedness. For a foliation \(\mathcal{F}\) on a smooth surface \(S\), locally generated by a vector field \(v\), the foliation is \(p\)-closed if \(v\wedge v^p=0\); if it is not \(p\)-closed, one defines the \(p\)-divisor
\[
\Delta_{\mathcal{F}}=\operatorname{div}(v\wedge v^p)
\]
[2507.07277]. On \(\mathbb{P}^2_k\), if \(\mathcal{F}\) has degree \(d\), then
\[
\deg(\Delta_{\mathcal{F}})=p(d-1)+d+2,
\]
so for \(p=2\) one gets \(\deg(\Delta_{\mathcal{F}})=3d\) [2507.07277]. In the formulation used in the arithmetic study of Jouanolou foliations, the \(p\)-divisor is also written as
\[
\Delta_{\mathcal{F}}=\{\,i_{v_\omega^p}\omega=0\,\}\in \operatorname{Div}(\mathbb{P}^2),
\]
again with
\[
\deg \Delta_{\mathcal{F}}=p(d-1)+d+2
\]
[2305.06251].

A fundamental property is that invariant irreducible curves are detected by \(\Delta_{\mathcal{F}}\): if \(C\) is \(\mathcal{F}\)-invariant, then \(C\subset \operatorname{Supp}(\Delta_{\mathcal{F}})\); conversely, if a prime divisor occurs in \(\Delta_{\mathcal{F}}\) with multiplicity not divisible by \(p\), then it is invariant [2305.06251]. This makes \(\Delta_{\mathcal{F}}\) a positive-characteristic analog of a global detector for algebraic leaves.

For the Jouanolou foliation, the characteristic-\(2\) computation is especially explicit. On the chart \(z\neq 0\), with
\[
v=(x y^d-1)\partial_x-(x^d-y^{d+1})\partial_y,
\]
one has in characteristic \(2\)
\[
v^2=((d+1)x y^{2d}-y^d-dx^{d+1}y^{d-1})\partial_x+((d+1)y^{2d+1}-(2d+1)x^d y^d+d x^{d-1})\partial_y,
\]
and the local equation of the \(2\)-divisor is
\[
f(x,y)=y^{2d+1}+x^d y^d+x^{2d+1}y^{d-1}+x^{d-1}
\]
[2305.06251]. For odd \(d\), this divisor is irreducible; for even \(d\), the foliation is \(2\)-closed [2305.06251]. In the later refinement, the irreducibility input is used for all odd \(d\), yielding a clean odd-degree result without the earlier congruence restriction \(d\not\equiv 1\!\!\pmod 3\) [2507.07277].

## 4. Modulo \(2\) criteria and new proofs of non-algebraicity

A central theorem in the recent arithmetic approach states that if a foliation \(\mathcal{F}\) on \(\mathbb{P}^2_{\mathbb{C}}\) of degree \(d>1\) is defined over a number field, is not dicritical, has good reduction modulo \(2\), and its reduction to \(\overline{\mathbb{F}}_2\) has irreducible \(2\)-divisor \(\Delta\), then \(\mathcal{F}\) has no algebraic invariant curves [2507.07277]. The proof combines four ingredients: good reduction, the fact that reductions of invariant curves cannot become \(p\)-factors, irreducibility of \(\Delta\), and Carnicer’s degree bound
\[
\deg(C)\le d+2
\]
for invariant irreducible curves without dicritical singularities [2507.07277].

Applied to the Jouanolou foliation, this yields the statement: if \(d\) is odd, then the Jouanolou foliation \(J_d\) has no algebraic invariant curves [2507.07277]. The parity assumption enters precisely through irreducibility of the \(2\)-divisor in characteristic \(2\).

This result refines earlier arithmetic work. An earlier reduction-modulo-\(2\) argument had established non-algebraicity for odd degree under an extra congruence condition \(d\not\equiv 1 \pmod 3\) [2305.06251]. The later paper removes that extra congruence hypothesis and proves the odd-degree statement uniformly [2507.07277]. By contrast, the even-degree case is not settled by that method; no claim is made there that the \(2\)-divisor is irreducible for even \(d\), and the argument is not asserted to apply [2507.07277].

The arithmetic literature also studies the full structure of the \(p\)-divisor for Jouanolou foliations in characteristic \(p\). Under assumptions including \(p<d\), \(p\not\equiv 1\pmod 3\), and primality of \(d^2+d+1\), the \(p\)-divisor is shown either to be irreducible or to decompose as
\[
\Delta_{\mathcal{I}_d}=C+pR,
\]
where \((\mathcal{I}_d,C)\) is a special pair and the remaining components are not \(\mathcal{I}_d\)-invariant [2305.06251]. This formulation shows that positive characteristic does not merely approximate characteristic-zero behavior: it introduces genuinely new phenomena such as \(p\)-factors.

## 5. Automorphisms, special pairs, and geometric constraints

The Jouanolou foliation possesses a large automorphism group. If \(\operatorname{char}(K)\) does not divide \(d^2+d+1\), and \(\gamma\) is a generator of the \((d^2+d+1)\)-th roots of unity, then
\[
\Phi:[x:y:z]\mapsto [\gamma^{d^2+1}x:\gamma y:z]
\]
satisfies
\[
\Phi^*\omega_d=\gamma \omega_d,
\]
so \(\Phi\in \operatorname{Aut}(\mathcal{I}_d)\) and has order \(d^2+d+1\) [2305.06251]. These large cyclic symmetries constrain invariant curves and play a decisive role in arithmetic arguments.

One such constraint states that if \(\mathcal{F}\) is non-\(p\)-closed of degree \(d\), \(p<d\), and \(\Phi\) has order \(e\ge d^2\), then any \(\mathcal{F}\)-invariant irreducible curve \(C\) whose \(\Phi\)-orbit has size \(e\) must satisfy \(\deg C=1\) [2305.06251]. Combined with congruence obstructions coming from \(p\)-reduced singularities, this often rules out invariant curves altogether.

Another notion is that of a **special pair** \((\mathcal{F},C)\). For an invariant curve \(C=\{F=0\}\) of degree \(e\), there exists a unique \(3\)-form \(\beta_{\mathcal{F},C}\) such that
\[
dF\wedge \omega = F\left(\frac{e}{d+2}d\omega+i_R\beta_{\mathcal{F},C}\right),
\]
and the pair is called special if \(\beta_{\mathcal{F},C}=0\), equivalently
\[
(d+2)dF\wedge \omega=eF\,d\omega
\]
[2305.06251]. When \(\operatorname{char}(k)=p>2\), \(\mathcal{F}\) is non-\(p\)-closed, and the singularities are \(p\)-reduced, specialness forces
\[
\deg(C)\equiv d+2 \pmod p,
\]
and every singular point of \(\mathcal{F}\) lies on \(C\) with multiplicity \(2\) [2305.06251]. These congruence and multiplicity restrictions explain why the arithmetic theory can often classify possible invariant components of \(\Delta_{\mathcal{F}}\) very rigidly.

## 6. Degree \(2\): structural stability, dynamics, and harmonic currents

The degree-\(2\) Jouanolou foliation has attracted separate attention because its dynamics are unusually rigid and explicit. In one formulation, it is the projectivization of
\[
V=(y^2,z^2,x^2),
\]
equivalently of the \(1\)-form
\[
\omega=(z^3-x^2 y)\,dx+(x^3-y^2 z)\,dy+(y^3-x z^2)\,dz
\]
[2303.09313]. In affine coordinates \((u,v)=(x/z,y/z)\), the induced vector field is
\[
X=(v^2-u^3)\frac{\partial}{\partial u}+(1-u^2 v)\frac{\partial}{\partial v}
\]
[2303.09313].

This foliation is structurally stable on \(\mathbb{C}\mathbb{P}^2\): there exists a neighborhood in the moduli space \(\mathcal{F}_2\) of degree-\(2\) foliations such that every foliation in that neighborhood is topologically conjugate to it [2303.09313]. The proof uses a leafwise gradient flow \(W\), a transverse real-analytic curve
\[
B=\Pi(\{R\cdot V=0\}),
\]
uniform hyperbolicity of \(W\) away from \(B\cup \operatorname{sing}\), and a locally free affine-group action along leaves [2303.09313]. The Fatou set is exactly the forward \(W\)-saturation \(\operatorname{Att}_W(B)\), and it is a smooth, locally trivial disk bundle over \(B\) [2303.09313]. Moreover, \(B\) is biholomorphic to the Klein quartic
\[
xy^3+yz^3+zx^3=0,
\]
so the Fatou set is a fibration on the Klein quartic with fibers given by properly embedded holomorphic disks [2303.09313]. In particular, there is no dense leaf [2303.09313].

The degree-\(2\) case also admits an explicit description of transverse harmonic current dimension. For a singular holomorphic foliation \(\mathcal{F}\) on a compact Kähler surface with hyperbolic singularities and no foliated cycle, the transverse Hausdorff dimension of the unique harmonic current is
\[
\dim_H(T)=\frac{h_D}{|\lambda|},
\]
where \(h_D\) is the Furstenberg entropy and
\[
\lambda=\frac{T\cdot N_{\mathcal{F}}}{T\cdot T_{\mathcal{F}}}
\]
is the Lyapunov exponent [2503.09152]. On \(\mathbb{P}^2\), if \(\mathcal{F}\) has degree \(d\ge 2\), then
\[
\lambda=-\frac{d+2}{d-1},
\qquad
\dim_H(T)\le \frac{d-1}{d+2},
\]
so the harmonic current is singular with respect to Lebesgue measure on transversals [2503.09152].

For the Jouanolou foliation of degree \(2\), the singularities are hyperbolic, there is no foliated cycle, the holonomy pseudogroup on the pseudo-minimal set is discrete, and the leaf entropy satisfies \(h_L=1\); since discreteness implies \(h_D=h_L\), one gets
\[
h_D=1,\qquad \lambda=-4,\qquad \dim_H(T)=\frac14
\]
[2503.09152]. The same dimension conclusion holds for foliations on \(\mathbb{P}^2\) that are topologically conjugate to the degree-\(2\) Jouanolou foliation [2503.09152]. This shows that the Jouanolou foliation is not only an obstruction to algebraic integrability but also a rigid benchmark in the metric theory of foliated dynamics.

## 7. Role in Darboux–Jouanolou theory and broader significance

In the Darboux–Jouanolou framework over arbitrary fields, invariant hypersurfaces are encoded by cofactors. If \(\omega\) is a polynomial \(r\)-form and \(F\) is irreducible, invariance means
\[
\omega\wedge dF=F\,\Theta_F
\]
[2102.00520]. Sufficiently many such invariant hypersurfaces force a rational first integral or an integrating factor, with the linear algebra carried out over \(K(z^p)\) in characteristic \(p\) [2102.00520], [1902.09571]. For \(r=1\), \(n=2\), and characteristic \(0\), the sharp threshold is \(d(d+1)/2+2\) invariant curves for a rational first integral [2102.00520].

The Jouanolou foliation is therefore the standard counterweight to integrability theorems. It demonstrates that below such thresholds one can have complete absence of algebraic invariant curves [1902.09571], [2102.00520]. In characteristic \(0\), it is the paradigm for generic non-algebraicity; in characteristic \(p\), it becomes a testing ground for \(p\)-closedness, \(p\)-divisors, special pairs, and reduction arguments [2305.06251], [2507.07277].

Recent work further shows that the reduction-modulo-\(2\) method is algorithmic in practice. Given a foliation defined by a polynomial vector field with integral coefficients, one forms the coefficient ring \( \mathbb{Z}[\mathcal{F}] \), checks good reduction modulo \(2\), computes the \(2\)-divisor via
\[
\frac{v\wedge v^2}{\partial_x\wedge \partial_y}=A\cdot v^2(y)-B\cdot v^2(x),
\]
tests irreducibility, verifies non-dicriticality, and then applies the main criterion [2507.07277]. The same framework yields further explicit families with either no algebraic invariant curves or exactly one algebraic invariant curve, often the line at infinity [2507.07277]. This suggests that the Jouanolou foliation now functions not only as a classical example but also as a template for constructing and certifying non-algebraic foliations by arithmetic-computational means.

In summary, the Jouanolou foliation occupies a singular position in foliation theory. It is simultaneously a classical example of non-algebraicity on \(\mathbb{P}^2\), a benchmark for Darboux–Jouanolou integrability thresholds, a natural object for reduction-modulo-\(p\) analysis through the \(p\)-divisor, and, in degree \(2\), a dynamically rigid foliation with explicitly computable transverse harmonic-current dimension [2305.06251], [2507.07277], [2303.09313], [2503.09152].

Source: https://www.emergentmind.com/topics/jouanolou-foliation