Bruno ideal and the variety of centers for singular germs of vector fields

Abstract

Given a logarithmic analytic vector field $\partial$, we consider the formal ideal $B(\partial )$ defined by the collinearity locus of the semi-simple and nilpotent components of~$\partial$. Assuming that the eigenvalues of the linear part of $\partial$ satisfy the so-called Bruno arithmetic condition, we prove that $B(\partial )$ is in fact an analytic ideal. Moreover, $\partial$ is analytically normalizable when restricted to this ideal. As a consequence, the vanishing locus $V$ of $B(\partial )$ is an analytic variety, and the foliation defined by $\partial {|}_V$ is analytically linearizable.