# Valente Ramirez Garcia Luna

### First Position

Lebesgue Post-doc Fellow, Institut de Recherche Mathématique de Rennes### Dissertation

*Quadratic vector fields on the complex plane: rigidity and analytic invariants*

### Advisor

### Research Area

### Abstract

This work deals with generic quadratic vector fields on $\mathbb{C}^2$ and the holomorphic foliations that these vector fields define on $\mathbb{C}\mathrm{P}^2$. We assume that the extended foliation has non-degenerate singularities only and an invariant line at infinity.

The first part of the present work deals with the extended spectra of singularities. The extended spectra is the collection of the eigenvalues of the linearization of the vector field at each of the singular points in the affine part, together with the characteristic numbers (i.e. Camacho-Sad indices) of the singularities on the line at infinity. We discuss what are the relations among these numbers that every generic quadratic vector field is bound to satisfy. Moreover, we conclude that two generic quadratic vector fields are affine equivalent if and only if they have the same extended spectra of singularities.

In the second part we focus on the holonomy group at infinity. We show that two generic quadratic vector fields that have conjugate holonomy groups must be orbitally affine equivalent. In particular, this proves that generic quadratic vector fields exhibit the utmost rigidity property: two such vector fields are orbitally topologically equivalent if and only if they are orbitally affine equivalent.