The Isotropy Group of a Foliation: The Local Case.

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. [ABSTRACT FROM AUTHOR]