# Automorphisms of holomorphic foliations on rational surfaces

*2002*

#### Translator’s note

*This page is a translation into English of the following:*

Sánchez, PF. “Automorfismo de foliaciones holomorfas sobre superficies racionales.” *Pro Mathematica* **16** (2002), 47–59. revistas.pucp.edu.pe/index.php/promathematica/article/view/8182/8478.

*The translator (Tim Hosgood) takes full responsibility for any errors introduced, and claims no rights to any of the mathematical content herein.*

Version: `4da29e4`

In this work, we classify the holomorphic foliations with infinite automorphism group on a rational surface. As a consequence of this result, we prove that the automorphism group of a foliation of general type with singularities on a rational surface is finite.

# 1 Introduction

Schwarz proved that the automorphism group of a Riemann surface of genus greater than 2 is finite. Andreotti, in [1], generalised this result, proving that the group of bimeromorphisms of an algebraic variety of general type is finite, these being analogous to Riemann surfaces of genus greater than 2. In the case of algebraic surfaces, there is even a bound for this number, see [14].

In this work we first classify the holomorphic foliations with singularities on rational surfaces.

Let

We then prove a result analogous to that of Andreotti for holomorphic foliations on rational surfaces (surfaces bimeromorphic to the projective plane).

The automorphism group of a holomorphic foliation of general type with singularities on a rational surface is finite.

# 2 Preliminaries

For the basic notions of holomorphic foliations, we recommend the books [4], [11], and [2].
A holomorphic foliation *cotangent*, or *canonical*, *bundle*.
The set of singularities

Let *tangency* between

If

Now suppose that

We define the *Camacho–Sad index* at *Camacho–Sad index theorem* [5] says that
*reduced foliation* is a foliation *saddle node*.

Let *Kodaira dimension* of

The concept of Kodaira dimension for holomorphic foliations was introduced independently by L.G. Mendes and M. McQuillan, see [2], [13], and [12].

It has been proven that the Kodaira dimension is well defined and is a bimeromorphic invariant of

When the foliation has Kodaira dimension *general type*.
This terminology is justified since there exists a partial classification of foliations of Kodaira dimension less than

A *fibration* at *generic* fibre of *connected*.
If all the fibres are rational curves then the fibration *rational* fibration.

A foliation *Riccati* foliation if there exists a rational fibration whose fibres are transversal to

In [13], it was proven that Riccati foliations and rational fibrations have dimension at most

The *bimeromorphism* (resp. *automorphism*) *group* of the foliation

In the case of

The automorphism group of a holomorphic foliation on

*Proof*. By definition,

Suppose that a foliation

The following result shows that this example is general.

If

*Proof*. Suppose, for contradiction, that

To obtain a holomorphic vector field in the Lie algebra of

# 3 Automorphisms of holomorphic foliations on the projective plane

Let

\omega=P\mathrm{d}{x}+(Q-yP)\mathrm{d}{y} , whereP,Q\in\mathbb{C}[x-\frac{y^2}{2}] ;\omega=yp(y)\mathrm{d}{x}+(xp(y)-q(y))\mathrm{d}{y} , wherep,q\in\mathbb{C}[t] ;\omega=yp(y^m/x^n)\mathrm{d}{x}+xq(y^m/x^n)\mathrm{d}{y}

or

*Proof*. Recall that the automorphism group of

Any global holomorphic vector field

Firstly, suppose that the canonical Jordan form of

Now consider

The remaining case is when

Suppose that

Finally, assume that

Let

*Proof*. From the proof of the previous theorem we have that the foliation defined by

Now suppose that the foliation

Thus the geometric place of the tangencies of

Let

*Proof*. Since

Note that the group

Since

Note that the image of

In the proof of our principal result, we approximately follows the arguments of Brunella in [2].

*Proof*. *(Proof of Theorem A).*
If

X|_{\Gamma_n}=0 . If there is a fibreF (of a rational fibration ofM on\Gamma_n ) that is not invariant then we have that\operatorname{tang}({\mathcal{G}},F)>0 . Since{\mathcal{G}} only has zero divisors,T_{\mathcal{G}}\cdot F\geqslant 0 , but on the other handT_{\mathcal{G}}\cdot F=F^2-\operatorname{tang}({\mathcal{G}},F)=-\operatorname{tang}({\mathcal{G}},F)<0 . SoF is invariant under{\mathcal{G}} , and soX is tangent to a rational fibration. Since\operatorname{tang}({\mathcal{G}},F) is invariant under{\mathcal{F}} andX (see [6]), it must be a rational fibration or a Riccati foliation.X|_{\Gamma_n}\neq0 . ThenX has more than two zeros on\Gamma_n and, by the argument used in the previous case, the fibres that pass through these points are{\mathcal{G}} -invariant. Furthermore, the Poincaré–Hopf index ofX|_{\Gamma_n} agrees with the multiplicity of zeros ofX|_{\Gamma_n} . So we have two cases:- There are two
{\mathcal{G}} -invariant fibres, and there does not exist a singularity outside of\Gamma_n , and so both singularities on\Gamma_n are saddle nodes and, by the index theorem,\Gamma_n^2=0 , which is a contradiction. - If there is only one
{\mathcal{G}} -invariant fibreF , then the multiplicity ofX/_{\Gamma_n} at the singular point is2 . Now, ifX has no singularities onF outside of\Gamma_n , thenX/_F has multiplicity2 . Thus, again by the index theorem,\Gamma_n^2=0 , which is a contradiction.

- There are two

Resuming, we have a fibre

From here we apply a blow-up and two blow-downs to arrive at ^{1}
To finish, we apply Corollary 2.

*Proof*. *(Proof of Theorem B).*
By Theorem 3.3.1 in [13], we know that the Riccati foliations and rational fibrations have Kodaira dimension less than

## 3.1 Final commentary

This result that we have proven in this article for rational surfaces is valid for any complex surface, see [6] and [7].
In this latter article we also prove some results in dimension higher than

## 3.2 Thanks

This article is a summary of a part of my doctoral thesis [6] which was written under the direction of Professor Cesar Camacho and an article co-authored with Jorge Pereira, both of whom I thank for the many comments and suggestions.

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*[Trans.] The original has a figure here, showing this bimeromorphism between*↩︎\mathbb{P}_\mathbb{C}^1\times\mathbb{P}_\mathbb{C}^1 and\mathbb{P}_\mathbb{C}^2 , which I have not tried to reproduce.