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.

We formalize the concepts of holomorphic affine and projective structures along the leaves of holomorphic foliations by curves on complex manifolds. We show that many foliations admit such structures, we provide local normal forms for them at singular points of the foliation, and we prove some index formulae in the case where the ambient manifold is compact. As a consequence of these, we establish that a regular foliation of general type on a compact algebraic manifold of even dimension does not admit a foliated projective structure. Finally, we classify foliated affine and projective structures along regular foliations on compact complex surfaces.

Let $(\mathbb {D}^2,\mathscr {F},\{0\})$ be a singular holomorphic foliation on the unit bidisc $\mathbb {D}^2$ defined by the linear vector field

$$ \begin{align*} z \frac{\partial}{\partial z}+ \unicode{x3bb} w \frac{\partial}{\partial w}, \end{align*} $$

where $\unicode{x3bb} \in \mathbb {C}^*$ . Such a foliation has a non-degenerate singularity at the origin ${0:=(0,0) \in \mathbb {C}^2}$ . Let T be a harmonic current directed by $\mathscr {F}$ which does not give mass to any of the two separatrices $(z=0)$ and $(w=0)$ . Assume $T\neq 0$ . The Lelong number of T at $0$ describes the mass distribution on the foliated space. In 2014 Nguyên (see [16]) proved that when $\unicode{x3bb} \notin \mathbb {R}$ , that is, when $0$ is a hyperbolic singularity, the Lelong number at $0$ vanishes. Suppose the trivial extension $\tilde {T}$ across $0$ is $dd^c$ -closed. For the non-hyperbolic case $\unicode{x3bb} \in \mathbb {R}^*$ , we prove that the Lelong number at $0$ :

1. (1) is strictly positive if $\unicode{x3bb}>0$ ;

2. (2) vanishes if $\unicode{x3bb} \in \mathbb {Q}_{<0}$ ;

3. (3) vanishes if $\unicode{x3bb} <0$ and T is invariant under the action of some cofinite subgroup of the monodromy group.

In this note we prove the following surprising characterization: if $X\,\subset \,{{\mathbb{A}}^{n}}$ is an (embedded, non-empty, proper) algebraic variety deûned over a field $k$ of characteristic zero, then $X$ is a hypersurface if and only if the module ${{T}_{{{O}_{{{\mathbb{A}}^{n\,/k}}}}}}(X)$ of logarithmic vector fields of $X$ is a reflexive ${{O}_{{{\mathbb{A}}^{n}}}}$-module. As a consequence of this result, we derive that if ${{T}_{{{O}_{{{\mathbb{A}}^{n\,/k}}}}}}(X)$ is a free ${{O}_{{{\mathbb{A}}^{n}}}}$-module, which is shown to be equivalent to the freeness of the $t$-th exterior power of ${{T}_{{{O}_{{{\mathbb{A}}^{n\,/k}}}}}}(X)$ for some (in fact, any) $t\,\le \,n$, then necessarily $X$ is a Saito free divisor.

In this paper we study the classification of holomorphic flows on Stein spaces of dimension two. We assume that the flow has periodic orbits, not necessarily with a same period. Then we prove a linearization result for the flow, under some natural conditions on the surface.

We study the infinite-dimensional group of holomorphic diffeomorphisms of certain Stein homogeneous spaces. We show that holomorphic automorphisms can be approximated by generalized shears arising from unipotent subgroups. For the homogeneous spaces this implies the existence of Fatou–Bieberbach domains of the first and second kind, and the failure of the Abhyankar–Moh theorem for holomorphic embeddings.