Holomorphic vector fields with totally degenerate zeroes

A zero set of a holomorphic vector field is totally degenerate, if the endomorphism of the conormal sheaf induced by the vector field is identically zero. By studying a class of foliations generalizing foliations of C^*-actions, we show that if a projective manifold admits a holomorphic vector field with a smooth totally degenerate zero component,then the manifold is stably birational to that component of the zero set.When the vector field has an isolated totally degenerate zero, we prove that the manifold is rational. This is a special case of Carrell's conjecture.