The Isotropy Group of a Foliation: The Local Case

Abstract

Abstract Given a holomorphic singular foliation ${\mathcal {F}}$ of $({\mathbb {C}}^n,0)$, we define $\textrm {Iso}({\mathcal {F}})$ as the group of germs of biholomorphisms on $({\mathbb {C}}^n,0)$ preserving ${\mathcal {F}}$: $\textrm {Iso}({\mathcal {F}})\!=\!\lbrace \Phi \in \textrm {Diff}({\mathbb {C}}^n,0)\,|\,\Phi ^*({\mathcal {F}})\!=\!{\mathcal {F}}\rbrace $. The normal subgroup of $\textrm {Iso}({\mathcal {F}})$, of biholomorphisms sending each leaf of ${\mathcal {F}}$ into itself, will be denoted as $\textrm {Fix}({\mathcal {F}})$. The corresponding groups of formal biholomorphisms will be denoted as $\widehat {\textrm {Iso}}({\mathcal {F}})$ and $\widehat {\textrm {Fix}}({\mathcal {F}})$, respectively. The purpose of this paper will be to study the quotients $\textrm {Iso}({\mathcal {F}})/\textrm {Fix}({\mathcal {F}})$ and $\widehat {\textrm {Iso}}({\mathcal {F}})/\widehat {\textrm {Fix}}({\mathcal {F}})$, mainly in the case of codimension one foliation.