Radial Foliations in Dimension Three

Radial germs of holomorphic foliations in dimension two have a characteristic property: they are the only singular foliations whose reduction of singularities has no singular points. We also know that they are desingularized by a single dicritical blowing-up. Let us say that a foliated space ((C³,0),E,F) ( ( C 3 , 0 ) , E , F ) is almost radial when it has a reduction of singularities without singular points; it will be “radial” under a certain additional condition on the morphism of reduction of singularities. We show that the radial condition corresponds to the “open book” situation. We end the paper with a discussion on the general almost radial case.