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

TL;DR

This paper proves that for certain logarithmic vector fields satisfying Bruno's arithmetic condition, the associated collinearity locus is an analytic variety where the vector field is analytically normalizable and linearizable.

Contribution

It establishes that under Bruno's condition, the collinearity locus of the vector field is an analytic variety and the vector field is analytically normalizable on this locus.

Findings

The ideal $B( ext{partial})$ is analytic under Bruno's condition.

The vector field is analytically normalizable on the vanishing locus of $B( ext{partial})$.

The foliation on this locus is analytically linearizable.

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.