Jouanolou Foliation on Complex Projective Planes

• Jouanolou foliation is a holomorphic foliation on the complex projective plane defined by explicit homogeneous 1-forms and marked by the absence of algebraic invariant curves for degrees greater than one.
• Arithmetic reduction modulo p and p-divisor analysis are employed to demonstrate that, for odd degrees, invariant curves cannot exist, underscoring its role as a counterexample in integrability theory.
• In its degree-2 form, the Jouanolou foliation exhibits structural stability and rich dynamics, with explicitly computed harmonic current dimensions highlighting its dynamic rigidity.

Searching arXiv for 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 $\mathbb{P}^2$ (Figueredo et al., 9 Jul 2025, Mendson, 2023). 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 (Mendson, 2023). Second, it has become a focal example linking complex foliation theory, arithmetic reduction modulo $p$, positive-characteristic $d$0-divisors, holonomy, and dynamical invariants such as harmonic currents (Figueredo et al., 9 Jul 2025, Deroin et al., 12 Mar 2025).

1. Definition and projective realization

A holomorphic foliation by curves of degree $d$1 on $d$2 can be represented by a homogeneous integrable $d$3-form

$d$4

satisfying the Euler condition $d$5 and the Frobenius integrability condition $d$6, where $d$7 is the radial vector field (Figueredo et al., 9 Jul 2025). In homogeneous coordinates $d$8, one writes

$d$9

with $$\omega_d=(x^{d} z-y^{d+1})\,dx+(x y^{d}-z^{d+1})\,dy+(z^{d} y-x^{d+1})\,dz,$$0 homogeneous of degree $$\omega_d=(x^{d} z-y^{d+1})\,dx+(x y^{d}-z^{d+1})\,dy+(z^{d} y-x^{d+1})\,dz,$$1 and $$\omega_d=(x^{d} z-y^{d+1})\,dx+(x y^{d}-z^{d+1})\,dy+(z^{d} y-x^{d+1})\,dz,$$2 (Mendson, 2023).

For the Jouanolou foliation of degree $$\omega_d=(x^{d} z-y^{d+1})\,dx+(x y^{d}-z^{d+1})\,dy+(z^{d} y-x^{d+1})\,dz,$$3, the affine vector field on the chart $$\omega_d=(x^{d} z-y^{d+1})\,dx+(x y^{d}-z^{d+1})\,dy+(z^{d} y-x^{d+1})\,dz,$$4 is

$$\omega_d=(x^{d} z-y^{d+1})\,dx+(x y^{d}-z^{d+1})\,dy+(z^{d} y-x^{d+1})\,dz,$$5

with associated affine $$\omega_d=(x^{d} z-y^{d+1})\,dx+(x y^{d}-z^{d+1})\,dy+(z^{d} y-x^{d+1})\,dz,$$6-form

$$\omega_d=(x^{d} z-y^{d+1})\,dx+(x y^{d}-z^{d+1})\,dy+(z^{d} y-x^{d+1})\,dz,$$7

A homogeneous extension to $$\omega_d=(x^{d} z-y^{d+1})\,dx+(x y^{d}-z^{d+1})\,dy+(z^{d} y-x^{d+1})\,dz,$$8 is

$$\omega_d=(x^{d} z-y^{d+1})\,dx+(x y^{d}-z^{d+1})\,dy+(z^{d} y-x^{d+1})\,dz,$$9

which satisfies $z\neq 0$0 and restricts to $z\neq 0$1 on $z\neq 0$2 (Figueredo et al., 9 Jul 2025). Equivalent projective formulas also appear in the arithmetic literature under the notation

$z\neq 0$3

(Mendson, 2023).

A closely related degree-$z\neq 0$4 model is given by the homogeneous vector field

$z\neq 0$5

whose projectivization defines the degree-$z\neq 0$6 Jouanolou foliation; the corresponding $z\neq 0$7-form is

$z\neq 0$8

(Alvarez et al., 2023). This suggests that the literature uses several equivalent normal forms, depending on whether one emphasizes affine vector fields, homogeneous $z\neq 0$9-forms, or projectivized polynomial vector fields.

2. Algebraic invariant curves and non-algebraicity

An irreducible algebraic curve $(x,y)$0, defined by a homogeneous polynomial $(x,y)$1, is invariant by a foliation $(x,y)$2 defined by $(x,y)$3 if and only if

$(x,y)$4

for some homogeneous $(x,y)$5-form $(x,y)$6; equivalently, $(x,y)$7 divides $(x,y)$8 in the homogeneous coordinate ring (Figueredo et al., 9 Jul 2025). In affine language, this is the standard tangency criterion for an invariant algebraic leaf.

Jouanolou’s classical theorem asserts that over $(x,y)$9, the Jouanolou foliation $v_d=(x y^d-1)\partial_x-(x^d-y^{d+1})\partial_y,$0 has no algebraic solutions for every $v_d=(x y^d-1)\partial_x-(x^d-y^{d+1})\partial_y,$1 (Mendson, 2023). 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 (Santos et al., 2019, Santos et al., 2021).

The contrast with Darboux–Jouanolou integrability is precise. For a polynomial $v_d=(x y^d-1)\partial_x-(x^d-y^{d+1})\partial_y,$2-form $v_d=(x y^d-1)\partial_x-(x^d-y^{d+1})\partial_y,$3, sufficiently many invariant irreducible hypersurfaces force a rational first integral. In characteristic $v_d=(x y^d-1)\partial_x-(x^d-y^{d+1})\partial_y,$4, for $v_d=(x y^d-1)\partial_x-(x^d-y^{d+1})\partial_y,$5 and $v_d=(x y^d-1)\partial_x-(x^d-y^{d+1})\partial_y,$6, the threshold is $v_d=(x y^d-1)\partial_x-(x^d-y^{d+1})\partial_y,$7 invariant curves for a rational first integral (Santos et al., 2021). The Jouanolou foliation lies at the opposite extreme: it has no algebraic invariant curve at all (Mendson, 2023). 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 $v_d=(x y^d-1)\partial_x-(x^d-y^{d+1})\partial_y,$8-divisor

A major recent development is the arithmetic study of the Jouanolou foliation via reduction modulo $v_d=(x y^d-1)\partial_x-(x^d-y^{d+1})\partial_y,$9. If $d$0 is defined over a finitely generated $d$1-algebra $d$2, one can reduce its defining coefficients modulo maximal ideals $d$3 and obtain a foliation $d$4 over a finite field of characteristic $d$5 (Figueredo et al., 9 Jul 2025, Mendson, 2023). This provides a bridge between complex foliations and positive-characteristic foliation theory.

In characteristic $d$6, one considers $d$7-closedness. For a foliation $d$8 on a smooth surface $d$9, locally generated by a vector field $\mathbb{P}^2$0, the foliation is $\mathbb{P}^2$1-closed if $\mathbb{P}^2$2; if it is not $\mathbb{P}^2$3-closed, one defines the $\mathbb{P}^2$4-divisor

$\mathbb{P}^2$5

(Figueredo et al., 9 Jul 2025). On $\mathbb{P}^2$6, if $\mathbb{P}^2$7 has degree $\mathbb{P}^2$8, then

$\mathbb{P}^2$9

so for $d>1$0 one gets $d>1$1 (Figueredo et al., 9 Jul 2025). In the formulation used in the arithmetic study of Jouanolou foliations, the $d>1$2-divisor is also written as

$d>1$3

again with

$d>1$4

(Mendson, 2023).

A fundamental property is that invariant irreducible curves are detected by $d>1$5: if $d>1$6 is $d>1$7-invariant, then $d>1$8; conversely, if a prime divisor occurs in $d>1$9 with multiplicity not divisible by $p$0, then it is invariant (Mendson, 2023). This makes $p$1 a positive-characteristic analog of a global detector for algebraic leaves.

For the Jouanolou foliation, the characteristic-$p$2 computation is especially explicit. On the chart $p$3, with

$p$4

one has in characteristic $p$5

$p$6

and the local equation of the $p$7-divisor is

$p$8

(Mendson, 2023). For odd $p$9, this divisor is irreducible; for even $d$00, the foliation is $d$01-closed (Mendson, 2023). In the later refinement, the irreducibility input is used for all odd $d$02, yielding a clean odd-degree result without the earlier congruence restriction $d$03 (Figueredo et al., 9 Jul 2025).

4. Modulo $d$04 criteria and new proofs of non-algebraicity

A central theorem in the recent arithmetic approach states that if a foliation $d$05 on $d$06 of degree $d$07 is defined over a number field, is not dicritical, has good reduction modulo $d$08, and its reduction to $d$09 has irreducible $d$10-divisor $d$11, then $d$12 has no algebraic invariant curves (Figueredo et al., 9 Jul 2025). The proof combines four ingredients: good reduction, the fact that reductions of invariant curves cannot become $d$13-factors, irreducibility of $d$14, and Carnicer’s degree bound

$d$15

for invariant irreducible curves without dicritical singularities (Figueredo et al., 9 Jul 2025).

Applied to the Jouanolou foliation, this yields the statement: if $d$16 is odd, then the Jouanolou foliation $d$17 has no algebraic invariant curves (Figueredo et al., 9 Jul 2025). The parity assumption enters precisely through irreducibility of the $d$18-divisor in characteristic $d$19.

This result refines earlier arithmetic work. An earlier reduction-modulo-$d$20 argument had established non-algebraicity for odd degree under an extra congruence condition $d$21 (Mendson, 2023). The later paper removes that extra congruence hypothesis and proves the odd-degree statement uniformly (Figueredo et al., 9 Jul 2025). By contrast, the even-degree case is not settled by that method; no claim is made there that the $d$22-divisor is irreducible for even $d$23, and the argument is not asserted to apply (Figueredo et al., 9 Jul 2025).

The arithmetic literature also studies the full structure of the $d$24-divisor for Jouanolou foliations in characteristic $d$25. Under assumptions including $d$26, $d$27, and primality of $d$28, the $d$29-divisor is shown either to be irreducible or to decompose as

$d$30

where $d$31 is a special pair and the remaining components are not $d$32-invariant (Mendson, 2023). This formulation shows that positive characteristic does not merely approximate characteristic-zero behavior: it introduces genuinely new phenomena such as $d$33-factors.

5. Automorphisms, special pairs, and geometric constraints

The Jouanolou foliation possesses a large automorphism group. If $d$34 does not divide $d$35, and $d$36 is a generator of the $d$37-th roots of unity, then

$d$38

satisfies

$d$39

so $d$40 and has order $d$41 (Mendson, 2023). These large cyclic symmetries constrain invariant curves and play a decisive role in arithmetic arguments.

One such constraint states that if $d$42 is non-$d$43-closed of degree $d$44, $d$45, and $d$46 has order $d$47, then any $d$48-invariant irreducible curve $d$49 whose $d$50-orbit has size $d$51 must satisfy $d$52 (Mendson, 2023). Combined with congruence obstructions coming from $d$53-reduced singularities, this often rules out invariant curves altogether.

Another notion is that of a special pair $d$54. For an invariant curve $d$55 of degree $d$56, there exists a unique $d$57-form $d$58 such that

$d$59

and the pair is called special if $d$60, equivalently

$d$61

(Mendson, 2023). When $d$62, $d$63 is non-$d$64-closed, and the singularities are $d$65-reduced, specialness forces

$d$66

and every singular point of $d$67 lies on $d$68 with multiplicity $d$69 (Mendson, 2023). These congruence and multiplicity restrictions explain why the arithmetic theory can often classify possible invariant components of $d$70 very rigidly.

6. Degree $d$71: structural stability, dynamics, and harmonic currents

The degree-$d$72 Jouanolou foliation has attracted separate attention because its dynamics are unusually rigid and explicit. In one formulation, it is the projectivization of

$d$73

equivalently of the $d$74-form

$d$75

(Alvarez et al., 2023). In affine coordinates $d$76, the induced vector field is

$d$77

(Alvarez et al., 2023).

This foliation is structurally stable on $d$78: there exists a neighborhood in the moduli space $d$79 of degree-$d$80 foliations such that every foliation in that neighborhood is topologically conjugate to it (Alvarez et al., 2023). The proof uses a leafwise gradient flow $d$81, a transverse real-analytic curve

$d$82

uniform hyperbolicity of $d$83 away from $d$84, and a locally free affine-group action along leaves (Alvarez et al., 2023). The Fatou set is exactly the forward $d$85-saturation $d$86, and it is a smooth, locally trivial disk bundle over $d$87 (Alvarez et al., 2023). Moreover, $d$88 is biholomorphic to the Klein quartic

$d$89

so the Fatou set is a fibration on the Klein quartic with fibers given by properly embedded holomorphic disks (Alvarez et al., 2023). In particular, there is no dense leaf (Alvarez et al., 2023).

The degree-$d$90 case also admits an explicit description of transverse harmonic current dimension. For a singular holomorphic foliation $d$91 on a compact Kähler surface with hyperbolic singularities and no foliated cycle, the transverse Hausdorff dimension of the unique harmonic current is

$d$92

where $d$93 is the Furstenberg entropy and

$d$94

is the Lyapunov exponent (Deroin et al., 12 Mar 2025). On $d$95, if $d$96 has degree $d$97, then

$d$98

so the harmonic current is singular with respect to Lebesgue measure on transversals (Deroin et al., 12 Mar 2025).

For the Jouanolou foliation of degree $d$99, the singularities are hyperbolic, there is no foliated cycle, the holonomy pseudogroup on the pseudo-minimal set is discrete, and the leaf entropy satisfies $$\omega_d=(x^{d} z-y^{d+1})\,dx+(x y^{d}-z^{d+1})\,dy+(z^{d} y-x^{d+1})\,dz,$$00; since discreteness implies $$\omega_d=(x^{d} z-y^{d+1})\,dx+(x y^{d}-z^{d+1})\,dy+(z^{d} y-x^{d+1})\,dz,$$01, one gets

$$\omega_d=(x^{d} z-y^{d+1})\,dx+(x y^{d}-z^{d+1})\,dy+(z^{d} y-x^{d+1})\,dz,$$02

(Deroin et al., 12 Mar 2025). The same dimension conclusion holds for foliations on $$\omega_d=(x^{d} z-y^{d+1})\,dx+(x y^{d}-z^{d+1})\,dy+(z^{d} y-x^{d+1})\,dz,$$03 that are topologically conjugate to the degree-$$\omega_d=(x^{d} z-y^{d+1})\,dx+(x y^{d}-z^{d+1})\,dy+(z^{d} y-x^{d+1})\,dz,$$04 Jouanolou foliation (Deroin et al., 12 Mar 2025). 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 $$\omega_d=(x^{d} z-y^{d+1})\,dx+(x y^{d}-z^{d+1})\,dy+(z^{d} y-x^{d+1})\,dz,$$05 is a polynomial $$\omega_d=(x^{d} z-y^{d+1})\,dx+(x y^{d}-z^{d+1})\,dy+(z^{d} y-x^{d+1})\,dz,$$06-form and $$\omega_d=(x^{d} z-y^{d+1})\,dx+(x y^{d}-z^{d+1})\,dy+(z^{d} y-x^{d+1})\,dz,$$07 is irreducible, invariance means

$$\omega_d=(x^{d} z-y^{d+1})\,dx+(x y^{d}-z^{d+1})\,dy+(z^{d} y-x^{d+1})\,dz,$$08

(Santos et al., 2021). Sufficiently many such invariant hypersurfaces force a rational first integral or an integrating factor, with the linear algebra carried out over $$\omega_d=(x^{d} z-y^{d+1})\,dx+(x y^{d}-z^{d+1})\,dy+(z^{d} y-x^{d+1})\,dz,$$09 in characteristic $$\omega_d=(x^{d} z-y^{d+1})\,dx+(x y^{d}-z^{d+1})\,dy+(z^{d} y-x^{d+1})\,dz,$$10 (Santos et al., 2021, Santos et al., 2019). For $$\omega_d=(x^{d} z-y^{d+1})\,dx+(x y^{d}-z^{d+1})\,dy+(z^{d} y-x^{d+1})\,dz,$$11, $$\omega_d=(x^{d} z-y^{d+1})\,dx+(x y^{d}-z^{d+1})\,dy+(z^{d} y-x^{d+1})\,dz,$$12, and characteristic $$\omega_d=(x^{d} z-y^{d+1})\,dx+(x y^{d}-z^{d+1})\,dy+(z^{d} y-x^{d+1})\,dz,$$13, the sharp threshold is $$\omega_d=(x^{d} z-y^{d+1})\,dx+(x y^{d}-z^{d+1})\,dy+(z^{d} y-x^{d+1})\,dz,$$14 invariant curves for a rational first integral (Santos et al., 2021).

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 (Santos et al., 2019, Santos et al., 2021). In characteristic $$\omega_d=(x^{d} z-y^{d+1})\,dx+(x y^{d}-z^{d+1})\,dy+(z^{d} y-x^{d+1})\,dz,$$15, it is the paradigm for generic non-algebraicity; in characteristic $$\omega_d=(x^{d} z-y^{d+1})\,dx+(x y^{d}-z^{d+1})\,dy+(z^{d} y-x^{d+1})\,dz,$$16, it becomes a testing ground for $$\omega_d=(x^{d} z-y^{d+1})\,dx+(x y^{d}-z^{d+1})\,dy+(z^{d} y-x^{d+1})\,dz,$$17-closedness, $$\omega_d=(x^{d} z-y^{d+1})\,dx+(x y^{d}-z^{d+1})\,dy+(z^{d} y-x^{d+1})\,dz,$$18-divisors, special pairs, and reduction arguments (Mendson, 2023, Figueredo et al., 9 Jul 2025).

Recent work further shows that the reduction-modulo-$$\omega_d=(x^{d} z-y^{d+1})\,dx+(x y^{d}-z^{d+1})\,dy+(z^{d} y-x^{d+1})\,dz,$$19 method is algorithmic in practice. Given a foliation defined by a polynomial vector field with integral coefficients, one forms the coefficient ring $$\omega_d=(x^{d} z-y^{d+1})\,dx+(x y^{d}-z^{d+1})\,dy+(z^{d} y-x^{d+1})\,dz,$$20, checks good reduction modulo $$\omega_d=(x^{d} z-y^{d+1})\,dx+(x y^{d}-z^{d+1})\,dy+(z^{d} y-x^{d+1})\,dz,$$21, computes the $$\omega_d=(x^{d} z-y^{d+1})\,dx+(x y^{d}-z^{d+1})\,dy+(z^{d} y-x^{d+1})\,dz,$$22-divisor via

$$\omega_d=(x^{d} z-y^{d+1})\,dx+(x y^{d}-z^{d+1})\,dy+(z^{d} y-x^{d+1})\,dz,$$23

tests irreducibility, verifies non-dicriticality, and then applies the main criterion (Figueredo et al., 9 Jul 2025). The same framework yields further explicit families with either no algebraic invariant curves or exactly one algebraic invariant curve, often the line at infinity (Figueredo et al., 9 Jul 2025). 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 $$\omega_d=(x^{d} z-y^{d+1})\,dx+(x y^{d}-z^{d+1})\,dy+(z^{d} y-x^{d+1})\,dz,$$24, a benchmark for Darboux–Jouanolou integrability thresholds, a natural object for reduction-modulo-$$\omega_d=(x^{d} z-y^{d+1})\,dx+(x y^{d}-z^{d+1})\,dy+(z^{d} y-x^{d+1})\,dz,$$25 analysis through the $$\omega_d=(x^{d} z-y^{d+1})\,dx+(x y^{d}-z^{d+1})\,dy+(z^{d} y-x^{d+1})\,dz,$$26-divisor, and, in degree $$\omega_d=(x^{d} z-y^{d+1})\,dx+(x y^{d}-z^{d+1})\,dy+(z^{d} y-x^{d+1})\,dz,$$27, a dynamically rigid foliation with explicitly computable transverse harmonic-current dimension (Mendson, 2023, Figueredo et al., 9 Jul 2025, Alvarez et al., 2023, Deroin et al., 12 Mar 2025).