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The spectra of singularities of quadratic vector fields

Le : 30/01/2018 16h45
Par : Valente Ramirez (Rennes)
Lieu : I 001
Lien web :
Résumé : Consider a polynomial vector field of degree $n \geq 2$ on $\mathbb{C}^2$. In the generic case, it has $n^2$ isolated singularities, and the holomorphic foliation it defines on $\mathbb{P}^2$ has an invariant line at infinity with $n+1$ singular points. Each equilibrium carries two numerical analytic invariants: the spectrum of its linearization matrix. Each singular point at infinity carries only one invariant: its Camacho-Sad index. Define the extended spectra of singularities to be the collection of these $2n^2+n+1$ numbers. These numbers are constrained by several index theorems, for example the Baum-Bott and the Camacho-Sad theorems. A dimensional argument shows that, for each fixed degree $n$, there must exist more algebraic relations among these numbers than the ones currently known. In this talk we will discuss the case of quadratic vector fields and describe all the relations among these numbers. Besides Baum-Bott, Camacho-Sad and the Euler-Jacobi relations, there is one more "hidden" relation. We will show how to obtain the hidden relation and explain its geometric significance.