César Camacho | Searcharxiv

arXiv2026

Geometric index theorems for holomorphic foliations and curves

Let $M$ be a complex surface endowed with a holomorphic projective connection and let $C \subset M$ be a compact smooth holomorphic curve. Given two singular holomorphic foliations $\mathcal{F}$ and $\mathcal{G}$ near $C$, with $C$ not $\mathcal{G}$-invariant, we attach to every point $p \in C$ an index $\mathrm{Ind}(\mathcal{F},\mathcal{G},C,p) \in \mathbb{C}$ of $\mathcal{F}$ relative to the reference $\mathcal{G}$, and we prove that these indices add up to $T\mathcal{G} \cdot C$. When $C$ is $\mathcal{F}$-invariant the index is the Camacho--Sad index corrected by the order of tangency of $\mathcal{G}$ with $C$, and the index formula reduces to the Camacho--Sad formula. The main applications are two contact formulas. At a tangency point $p$ of $\mathcal{G}$ with $C$, let $k(\mathcal{G},C,p)$ be the ratio of the curvatures at $p$ of the leaf of $\mathcal{G}$ and of the curve; it is a projective invariant, although each curvature separately depends on a choice of metric. If $\mathcal{G}$ has no singular points on $C$ and only simple tangencies with $C$, then $$ \sum_{p} \frac{1}{1 - k(\mathcal{G},C,p)} = \frac{2}{3}\left( C \cdot C + g - 1 \right), $$ where $g$ is the genus of $C$: the number of tangencies depends on $\mathcal{G}$, but this weighted count does not. For a pencil of lines in the projective plane it is the Plücker formula for the class of a plane curve. The second formula asserts that, for a curve in general position with respect to $\mathcal{F}$ and $\mathcal{G}$ which is not a geodesic, the total index of the tangencies of $\mathcal{F}$ with $\mathcal{G}$ along $C$ equals the number of tangencies of $\mathcal{G}$ with $C$ minus one third of the number of inflection points of $C$; in particular, it does not depend on $\mathcal{F}$.

math.CV↗

arXiv2021

Algebraic curves and foliations

Consider a field $k$ of characteristic $0$, not necessarily algebraically closed, and a fixed algebraic curve $f=0$ defined by a tame polynomial $f\in k[x,y]$ with only quasi-homogeneous singularities. We prove that the space of holomorphic foliations in the plane ${mathbb A}^2_\k$ having $f=0$ as a fixed invariant curve is generated as $k[x,y]$-module by at most four elements, three of them are the trivial foliations $fdx,fdy$ and $df$. Our proof is algorithmic and constructs the fourth foliation explicitly. Using Serre's GAGA and Quillen-Suslin theorem, we show that for a suitable field extension $K$ of $k$ such a module over $K[x,y]$ is actually generated by two elements, and therefore, such curves are free divisors in the sense of K. Saito. After performing Groebner basis for this module, we observe that in many well-known examples, $K=k$.

math.AG↗

arXiv2013

Invariant sets near singularities of holomorphic foliations

Consider a complex one dimensional foliation on a complex surface near a singularity $p$. If $\mathcal{I}$ is a closed invariant set containing the singularity $p$, then $\mathcal{I}$ contains either a separatrix at $p$ or an invariant real three dimensional manifold singular at $p$.

math.DS↗

arXiv2010

Extension theorems for analytic objects associated to foliations

In this paper we will establish a structure theorem concerning the extension of analytic objects associated to germs of dimension one foliations on surfaces, through one-dimensional barriers. As an application, an extension theorem for projective transverse structures is obtained.

math.CV↗