Preprint A266/2003
Curvature of pencils of foliations

Alcides Lins Neto

**Keywords: **
holomorphic foliations | pencil | curvature

\abstract{Let $\Cal{F}$ and $\Cal G$ be two distinct singular holomorphic foliations on a compact complex surface $M$, in the same class, that is $N_\Cal{F}=N_\Cal G$. In this case, we can define the {\it pencil $\Cal{P}=\Cal{P}(\Cal{F},\Cal G)$ of foliations generated by $\Cal{F}$ and} $\Cal G$. We can associate to a pencil $\Cal{P}$ a meromorphic 2-form $\Theta=\Theta(\Cal{P})$, the form of curvature of the pencil, which is in fact the Chern curvature (cf. [Ch]). When $\Theta(\Cal{P})\equiv 0$ we will say that the pencil is {\it flat}. In this paper we give some sufficient condictions for a pencil to be flat. (Theorem 2). We will see also how the flatness reflects in the pseudo-group of holonomy of the foliations of $\Cal{P}$. In particular, we will study the set
$\lbrace\Cal{H}\in \Cal{P}|\,\Cal{H} $ has a first integral $\rbrace$ in some cases (Theorem 1).}