# Singularities of holomorphic codimension one foliations of the complex projective plane

Dominique Cerveau, Julie Déserti

► To cite this version:

Dominique Cerveau, Julie Déserti. Singularities of holomorphic codimension one foliations of the complex projective plane. Boletim da Sociedade Brasileira de Matemática / Bulletin of the Brazilian Mathematical Society, 2023, 54 (2), pp.11. 10.1007/s00574-023-00332-0. hal-04335451

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Submitted on 11 Dec 2023

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# SINGULARITIES OF HOLOMORPHIC CODIMENSION ONE FOLIATIONS OF THE COMPLEX PROJECTIVE PLANE

DOMINIQUE CERVEAU AND JULIE DÉSERTI

**ABSTRACT.** We prove that any holomorphic codimension 1 foliation on the complex projective plane has at most one singularity up to the action of an ad-hoc birational map. Consequently, any algebraic foliation on the affine plane has no singularities up to the action of a suitable birational self map of the complex projective plane into itself.

Let $\mathcal{F}$ be a codimension one holomorphic foliation of degree $N$ on $\mathbb{P}_C^2$. Denote by $\pi: \mathbb{C}^3 \setminus \{0\} \to \mathbb{P}_C^2$ the canonical projection. The homogeneous foliation $\pi^{-1}\mathcal{F}$ extends to $\mathbb{C}^3$ and is defined by a 1-form

$$\omega = a(x, y, z) \, dx + b(x, y, z) \, dy + c(x, y, z) \, dz$$

where $a, b, c$ are homogeneous polynomials of degree $N+1$ without common component satisfying the Euler identity: $ax+by+cz=0$; it is the Chow theorem for foliations. The singular set $\text{Sing}(\mathcal{F})$ of $\mathcal{F}$ is given by

$$\pi(\{a=b=c=0\} \setminus \{0\}).$$

Let us recall what is the Milnor number $\mu(\mathcal{F}, m)$ of a foliation $\mathcal{F}$ at a singular point $m$. Let us fix a local chart $(u, v)$ such that $m = (0,0)$. The germ of $\mathcal{F}$ at $m$ is defined, up to multiplication by a unit at 0, by a 1-form $E \, du + F \, dv$. Denote by $\langle E, F \rangle$ the ideal generated by $E$ and $F$, then

$$\mu(\mathcal{F}, m) = \dim \frac{\mathbb{C}\{u, v\}}{\langle E, F \rangle};$$

it is also the multiplicity of intersection of the germs of curves $(E=0)$ and $(F=0)$. Let us recall that ([Bru15, Chapter 2, Section 3, Example 2])

$$\sum_{m \in \text{Sing}(\mathcal{F})} \mu(\mathcal{F}, m) = N^2 + N + 1;$$

in particular, $\mathcal{F}$ has at least one singular point.

<footnote n="2020">2020 Mathematics Subject Classification. 32S65.</footnote>

<footnote>Key words and phrases. Singularities of holomorphic foliations.</footnote>
**Example.** Let us consider the diagonal linear foliation $\mathcal{F}$ given by the 1-form

$$ \omega = \lambda yz dx + xz dy - (1 + \lambda)xy dz $$

with $\lambda \in \mathbb{C} \setminus \{0, -1, -2\}$. Note that $\mathrm{Sing}(\mathcal{F}) = \{(0:0:1), (1:0:0), (0:1:0)\}$; moreover, $x^{\lambda}yz^{-(1+\lambda)}$ is a first integral of $\mathcal{F}$.

The birational involution from $\mathbb{P}_C^2$ into itself given by

$$ \mathcal{I}_1: (x: y: z) \dashrightarrow (xy: y^2: x^2 - yz), $$

defines an isomorphism of $\{y \neq 0\}$. Remark that the foliation $\mathcal{I}_1^{-1}\mathcal{F}$ is described by the 1-form

$$ -y((2+\lambda)x^2 + \lambda yz) \, dx + x((2+\lambda)x^2 - yz) \, dy + (1+\lambda)xy^2 \, dz; $$

in particular, $\mathrm{Sing}(\mathcal{I}_1^{-1}\mathcal{F}) = \{(0:0:1), (0:1:0)\}$.

Let us now consider the birational involution $\mathcal{I}_2$ from $\mathbb{P}_C^2$ into itself given by

$$ \mathcal{I}_2: (x: y: z) \dashrightarrow (x^2: -xy + z^2: xz); $$

note that $\mathcal{I}_2$ defines an isomorphism of $\{x \neq 0\}$. Furthermore, the restriction of $\mathcal{I}_2^{-1}\mathcal{I}_1^{-1}\mathcal{F}$ on $(x=0)$ is described by the 1-form $(1-\lambda)z^5 \, dx$; consequently, $\mathrm{Sing}(\mathcal{I}_2^{-1}\mathcal{I}_1^{-1}\mathcal{F}) = \{(0:1:0)\}$.

Can we generalize this to all holomorphic foliations on $\mathbb{P}_C^2$? The answer is yes:

**Theorem 1.** Let $\mathcal{F}$ be a holomorphic foliation on $\mathbb{P}_C^2$. There exists a birational self map $\phi$ from $\mathbb{P}_C^2$ into itself such that $\phi^{-1}\mathcal{F}$ has at most one singular point.

Note that many properties are preserved by conjugacy by birational maps, for instance the existence of invariant algebraic curves of genus $\geq 2$, but also the existence of dense leaves.

**Corollary 2.** Let $\mathcal{F}$ be an algebraic foliation on $\mathbb{C}^2$. There exists a birational map $\phi$ from $\mathbb{C}^2$ into itself such that $\phi^{-1}\mathcal{F}$ has no singular point.

In a certain sense Corollary 2 indicates that algebraic foliations of $\mathbb{C}^2$ have no special properties.

Note that the situation is completely different on $\mathbb{P}_C^1 \times \mathbb{P}_C^1$; in that space there exist foliations without singularities: a foliation on $\mathbb{P}_C^1 \times \mathbb{P}_C^1$ birationally conjugate to a foliation without singularities has a rational first integral ([Bru15]).

To prove Theorem 1 we will use the following statement:

**Lemma 3.** Let $\mathcal{F}$ be a holomorphic codimension one foliation on $\mathbb{P}_C^2$. Assume that $\mathcal{F}$ has $N$ singular points, and among them $n \geq 2$ singular points on a line L. There exists a birational self map $\phi$ from $\mathbb{P}_C^2$ into itself such that $\phi^{-1}\mathcal{F}$ has $N-n+1$ singular points, one on L and $N-n$ on $\mathbb{P}_C^2 \setminus L$.
Proof. ◇ Assume first that $L$ is not invariant by $\mathcal{F}$. Let us choose coordinates such that

$$ egin{cases} L = (z = 0) \\ (0 : 1 : 0) \notin \operatorname{Sing}(\mathcal{F}) \\ L \text{ is not tangent to the leaf of } \mathcal{F} \text{ through } (0 : 1 : 0) \text{ at } (0 : 1 : 0) \quad (*) \end{cases} $$

The foliation $\mathcal{F}$ is defined by

$$ \omega = A(x, y, z) \, dx + B(x, y, z) \, dy + C(x, y, z) \, dz. $$

Let us recall the Euler condition: $xA + yB + zC = 0$; in particular note that $B(0, 1, 0) = 0$. The tangency condition (*) says that $A(0, 1, 0) \neq 0$.

Let us consider the birational involution $\phi$ of $\mathbb{P}_C^2$ into itself given by

$$ \phi: (x : y : z) \dashrightarrow (xz : -yz + x^2 : z^2); $$

note that

$$ \mathrm{Ind}(\phi) = \{(0 : 1 : 0)\}, \qquad \mathrm{Exc}(\phi) = \{(z = 0)\} $$

where $\mathrm{Ind}(\phi)$ (resp. $\mathrm{Exc}(\phi)$) denotes the set of indeterminacy (resp. the exceptional set) of $\phi$. In particular, $\phi$ defines an isomorphism of $\{z \neq 0\}$. One has

$$ \phi^* \omega = A(xz, -yz + x^2, z^2)(x \, dz + z \, dx) \\ + B(xz, -yz + x^2, z^2)(-y \, dz - z \, dy + 2x \, dx) + 2C(xz, -yz + x^2, z^2)z \, dz, $$

and

$$ \phi^* \omega \Big|_{z=0} = xA(0, x^2, 0) \, dx + \underbrace{B(0, x^2, 0)}_{=0} \, (-y \, dz + 2x \, dx) = xA(0, x^2, 0) \, dx; $$

in particular $\phi^* \omega|_{z=0}$ vanishes only at $x=0$ since $A(0, 1, 0) \neq 0$. As a consequence, $\#(\mathrm{Sing}(\phi^{-1}\mathcal{F}) \cap (z=0)) = 1$ and $\#\mathrm{Sing}(\phi^{-1}\mathcal{F}) = N-n+1$.

◇ Suppose now that $L$ is invariant by $\mathcal{F}$. Let us choose coordinates such that $\begin{cases} L = (z = 0) \\ (0 : 1 : 0) \notin \mathrm{Sing}(\mathcal{F}) \end{cases}$

The foliation $\mathcal{F}$ is defined by

$$ \omega = zA(x, y, z) \, dx + zB(x, y, z) \, dy + C(x, y, z) \, dz. $$

Let us still consider the birational involution $\phi$ of $\mathbb{P}_C^2$ into itself given by

$$ \phi: (x : y : z) \dashrightarrow (xz : -yz + x^2 : z^2) $$

that defines an isomorphism of $\{z \neq 0\}$. Then

$$ \phi^* \omega = z \left( \begin{aligned} & zA(xz, -yz + x^2, z^2)(x \, dz + z \, dx) \\ & + zB(xz, -yz + x^2, z^2)(-y \, dz - z \, dy + 2x \, dx) + 2C(xz, -yz + x^2, z^2) \, dz \end{aligned} \right), $$
and the restriction of $\phi^{-1}\mathcal{F}$ on ($z=0$) is described by the 1-form $2C(0,x^2,0)\,dz$. But $C(0,1,0) \neq 0$ (because $(0:1:0) \notin \text{Sing}(\mathcal{F})$) so $C(0,x^2,0) = 0$ if and only if $x=0$. In particular, $\#(\text{Sing}(\phi^{-1}\mathcal{F}) \cap (z=0)) = 1$ and $\#\text{Sing}(\phi^{-1}\mathcal{F}) = N-n+1$.

□

*Proof of Theorem 1.* We get the result by iteration using Lemma 3.

□

Assume that $\mathcal{F}$ is an algebraic foliation of any codimension of $\mathbb{C}^n$; is it true that there exists a birational self map of $\mathbb{P}_C^n$ into itself such that $\phi^{-1}\mathcal{F}$ has no singularity ? In the real case (for example in $\mathbb{P}_R^2$) what is the best result we can expect ?

Remark that following [CDGBM10] in which we classify, up to automorphisms of $\mathbb{P}_C^2$, the quadratic foliations having a unique singularity there is a lot of activity around the foliations of $\mathbb{P}_C^2$ having a unique singularity (for instance [FPR22, Alc22, Cou22, CJ21, CF21, CF20]).

## REFERENCES

[Alc22] C. R. Alcántara. Special foliations on $\mathbb{CP}^2$ with a unique singular point. *Res. Math. Sci.*, 9(1):Paper No. 15, 11, 2022.

[Bru15] M. Brunella. *Birational geometry of foliations*, volume 1 of IMPA Monographs. Springer, Cham, 2015.

[CDGBM10] D. Cerveau, J. Déserti, D. Garba Belko, and R. Meziani. Géométrie classique de certains feuilletages de degré deux. *Bull. Braz. Math. Soc.* (N.S.), 41(2):161–198, 2010.

[CF20] S. C. Coutinho and F. R. Ferreira. A family of foliations with one singularity. *Bull. Braz. Math. Soc.* (N.S.), 51(4):957–974, 2020.

[CF21] S. C. Coutinho and F. R. Ferreira. Correction to: A family of foliations with one singularity. *Bull. Braz. Math. Soc.* (N.S.), 52(3):767–769, 2021.

[CJ21] S. C. Coutinho and L. F. G. Jales. Foliations with one singularity and finite isotropy group. *Bull. Sci. Math.*, 169:Paper No. 102988, 18, 2021.

[Cou22] S. C. Coutinho. On the classification of foliations of degree three with one singularity. *J. Symbolic Comput.*, 112:62–78, 2022.

[FPR22] P. Fernández, L. Puchuri, and R. Rosas. Foliations on $\mathbb{P}^2$ with only one singular point. *Geom. Dedicata*, 216(5):Paper No. 59, 17, 2022.

UNIV. RENNES, CNRS, IRMAR-UMR 6625, F-35000 RENNES, FRANCE

Email address: dominique.cerveau@univ-rennes1.fr

UNIVERSITÉ D'ORLÉANS, INSTITUT DENIS POISSON, ROUTE DE CHARTRES, 45067 ORLÉANS CEDEX 2, FRANCE

Email address: deserti@math.cnrs.fr