Translator’s note

This page is a translation into English of the following:

Fernández Sánchez, P. “Automorfismo de foliaciones holomorfas sobre superficies racionales.” Pro Mathematica 16 (2002), 47–59. (Available online.)

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

Version: 7f60475

In this work we classify 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?], generalises this result, proving that the bimeromorphism group 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 TO-DO for this number, cf. [14?].

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

Let {\mathcal{F}} be a foliation on a rational surface M. If \#\operatorname{Aut}({\mathcal{F}})=+\infty, then {\mathcal{F}} is bimeromorphic to a Riccati foliation or a rational fibration.

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 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?,13?,2?]. A holomorphic foliation {\mathcal{F}} with isolated singularities on an algebraic surface M can be defined as a family \{X_i\} of holomorphic vector fields defined on an open cover \{U_i\} of M that satisfies the cocycle condition X_i=g_{ij}X_j whenever U_i\cap U_j\neq\varnothing, where \{g_{ij}\} are nowhere-zero holomorphic functions defined on U_i\cap U_j. In this case, \{g_{ij}\} defines a line bundle T_{{\mathcal{G}}}^* on M called the cotangent (or canonical) bundle. The set of singularities \operatorname{Sing}({\mathcal{G}}) of {\mathcal{G}} is defined as \operatorname{Sing}({\mathcal{G}})_{/U_i}=\{X_i=0\}, and it is always possible to suppose that \dim(\operatorname{Sing}({\mathcal{G}}))=0.

Let {\mathcal{F}} be a holomorphic foliation on a complex surface M. Consider a compact curve C on M. Let p\in C, and let \{f=0\} be a reduced local equation of C around p. Suppose that {\mathcal{F}} is represented in a coordinate neighbourhood (U,(x,y)) of p=(0,0) by the holomorphic 1-form \omega = a(x,y)\mathrm{d}x + b(x,y)\mathrm{d}y. If C is not F-invariant, then we define the tangency between {\mathcal{F}} and C at p by \operatorname{tang}_p({\mathcal{F}},C)=\dim_\mathbb{C}\mathscr{O}_p/I, where I is the ideal generated by f and -b\frac{\partial f}{\partial x}+a\frac{\partial f}{\partial y}. We define \operatorname{tang}({\mathcal{F}},C)=\sum_{p\in C}\operatorname{tang}_p({\mathcal{F}},C). It is proven in [2?] that T_{\mathcal{F}}^*C = \operatorname{tang}({\mathcal{F}},C) - C^2.

If M=\mathbb{C}P(2) and C is a non-{\mathcal{F}}-invariant line, then we have T_{\mathcal{F}}^* = \mathscr{O}_{\mathbb{C}P(2)}(\operatorname{tang}({\mathcal{F}},C)-1).

Now suppose that C is {\mathcal{F}}-invariant, so that \omega\wedge\mathrm{d}f=f\Theta, where \Theta is a holomorphic 2-form. Then there exist relatively prime holomorphic functions g and h defined on U, along with a holomorphic 1-form \eta, such that g\omega = h\mathrm{d}f + f\eta.

We define the Camacho–Sad index by \operatorname{CS}_p({\mathcal{F}},C)=-\frac{-1}{2\pi i}\int_\gamma\frac{\eta}{h}, where \gamma is a loop around p on \{f=0\}. We define \operatorname{CS}({\mathcal{F}},C)=\sum_{p\in\operatorname{Sing}{\mathcal{F}}\cap C}\operatorname{CS}_p({\mathcal{F}},C) The Camacho–Sad index theorem [5?] says that \operatorname{CS}({\mathcal{F}},C) = C^2. Recall that a reduced foliation {\mathcal{F}} is a foliation such that every singularity p is reduced in the sense of Seidenberg, i.e., for every vector field X generating {\mathcal{F}}, and for every singular point p of X, the eigenvalues of the linear part of X are not both zero, and their quotient, when defined, is not a positive rational number. If one eigenvalue is zero and the other is not, then the singularity is said to be a TO-DO: knot saddle?.

Let {\mathcal{F}} be a foliation on a complex surface S, and let {\mathcal{G}} be an arbitrary reduced foliation that is bimeromorphically equivalent to {\mathcal{F}}. The Kodaira dimension of {\mathcal{F}} is given by \operatorname{Kod}({\mathcal{F}}) = \limsup_{n\to\infty} \frac{\log h^0(S,K_{\mathcal{G}}^{\otimes=n})}{\log n}.