Related papers: Invariant sets near singularities of holomorphic f…
In this paper we study holomorphic foliations on $\mathbb{P}^2$ with only one singular point. If the singularity has algebraic multiplicity one, we prove that the foliation has no invariant algebraic curve. We also present several examples…
A holomorphic foliation is defined as an integrable coherent subsheaf of the tangent sheaf. The structure of the leaves around a singularity is read off from the structure of the stalks. This was done by Baum when the dimension of the…
The Separatrix Theorem of C. Camacho and P. Sad guarantees the existence of invariant curve (separatrix) passing through the singularity of germ of holomorphic foliation on complex surface, when the surface underlying the foliation is…
Let $\mathcal F$ be a holomorphic one-dimensional foliation on $\mathbb{P}^n$ such that the components of its singular locus $\Sigma$ are curves $C_i$ and points $p_j$. We determine the number of $p_j$, counted with multiplicities, in terms…
A singular foliation $\mathcal F$ gives a partition of a manifold $M$ into leaves whose dimension may vary. Associated to a singular foliation are two complexes, that of the diffeological differential forms on the leaf space $M / \mathcal…
We study codimension $q \geq 2$ holomorphic foliations defined in a neighborhood of a point $P$ of a complex manifold that are completely integrable, i.e. with $q$ independent meromorphic first integrals. We show that either $P$ is a…
This work concerns the problem of relating characteristic numbers of one-dimensional holomorphic foliations of P^n to those of algebraic varieties invariant by them. More precisely: if M is a connected complex manifold, a one-dimensional…
We study the classification of singularities of holomorphic foliations and non-integrable one-forms under the hypothesis of transversality with real hypersurfaces.
We define the Milnor number -- as the intersection number of two holomorphic sections -- of a one-dimensional holomorphic foliation $\mathscr{F}$ with respect to a compact connected component $C$ of its singular set. Under certain…
We consider singular holomorphic foliations on compact complex surfaces with invariant rational nodal curve of positive self-intersection. Then, under some assumptions, we list all possible foliations.
We review properties of closed meromorphic $1$-forms and of the foliations defined by them. We present and explain classical results from foliation theory, like index theorems, the existence of separatrices, and resolution of singularities…
Let X be a compact complex surface with a real foliation. If all leaves are compact complex curves, the foliation must be holomorphic.
It is known that there is at least an invariant analytic curve passing through each of the components in the complement of nodal singularities, after the reduction of singularities of a germ of singular foliation in ${\mathbb C}^2,0$}.…
Consider a holomorphic foliation with singularities of a 2-dimensional complex manifold. In this article we prove a new sufficient condition for this foliation to have countably many homologically independent complex limit cycles. In…
Let $\mathcal{F}$ be a singular Riemann surface foliation on a complex manifold $M$, such that the singular set $E \subset M$ is non-discrete. We study the behavior of the foliation near the singular set $E$, particularly focusing on…
In this expository article, we study and discuss invariants of vector fields and holomorphic foliations that intertwine the theories of complex analytic singular varieties and singular holomorphic foliations on complex manifolds: two…
We study the invariant theory of singular foliations of the projective plane. Our first main result is that a foliation of degree m>1 is not stable only if it has singularities in dimension 1 or contains an isolated singular point with…
We study codimension one holomorphic foliations on complex projective spaces and compact manifolds under the assumption that the foliation has a projective transverse structure in the complement of some invariant codimension one analytic…
We study the singular set of a codimension one holomorphic foliations on $\mathbb{P}^3$. We find a local normal form of a codimension two component of the singular set that is not of Kupka type. We also determined the number of non-Kupka…
A sequence of monoidal transformations is defined, in terms of invariants, for a singular hypersurface embedded in a smooth scheme of positive characteristic. Some examples are added to illustrate the improvement of singularities by this…