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

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