Preprint A183/2002
Exceptional families of foliations and the Poincare' problem
Alcides Lins Neto
Keywords: algebraic foliation | Poincare' problem
A 1-parameter family of foliations $(\fa_\a)_{\a\in X}$ on a compact complex surface $M$ is called {\it exceptional and elliptic} if it satisfies the following properties : {\bf (a).} The family has singularities of fixed analytic type; {\bf (b).} The set $E=\{\a\in X|\, \fa_\a$ has a first integral $\}$ is countable and non-discrete; {\bf (c).} There is $\a\in E$ such that the generic fibre of the first integral is elliptic. In this paper we show that, if a surface $M$ admits an exceptional and elliptic family of foliations, then $M$ is algebraic and biholomorphically equivalent to a torus, to a $K3$ surface, or to $\Cp(2)$ (Theorem 3). In the case of $\Cp(2)$ we classify all possible equireducible and exceptional families such that the singularities of the generic foliations in the family are non-degenerate (Theorem 2). This classification is connected to the Poincaré problem of deciding if an algebraic foliation on $\Cp(2)$ has a first integral (cf. [P-1]).