We investigate the geometry of codimension one foliations on smooth projective varieties defined over fields of positive characteristic with an eye toward applications to the structure of codimension one holomorphic foliations on projective manifolds.

A logarithmic 1-form on $\mathbb C\mathbb P^n$ can be written as

\omega=\biggl(\prod_0^m F_j\biggr)\sum_0^m \lambda_i\frac{dF_i}{F_i}=\lambda_0 \widehat F_0 \,dF_0+\cdots+\lambda_m \widehat F_m \,dF_m

with $\widehat F_i=(\prod_0^m F_j)/F_i$ for some homogeneous polynomials Fi of degree di and constants $\lambda_i\in{\mathbb C}^\star$ such that $\sum\lambda_id_i=0$. For general $F_i,\lambda_i$, the singularities of $\omega$ consist of a schematic union of the codimension 2 subvarieties Fi = Fj = 0 together with, possibly, finitely many isolated points. This is the case when all Fi are smooth and in general position. In this situation, we give a formula which prescribes the number of isolated singularities.

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 ${\rm C}^*$-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.