Simultaneous reduction to normal forms of commuting singular vector fields with linear parts having Jordan blocks

We study a simultaneous linearizability of d–actions (and the corresponding d-dimensional Lie algebras) defined by commuting singular vector fields in Cn fixing the origin with a nontrivial Jordan block in the linear parts. We prove the analytic convergence of a formal linearizing transformation under a certain invariant geometric condition (cone condition) for the spectrum of d vector fields generating a Lie algebra. (cf. Example 1.4.) If the condition fails, then we show the existence of divergent solutions of an overdetermined system of linearized homological equations. In a smooth category, the situation is ∗Supported by NATO grant PST.CLG.979347, and GNAMPA-INDAM, Italy. †Supported by Grant-in-Aid for Scientific Research (No. 14340042), Ministry of Education, Science and Culture, Japan and GNAMPA-INDAM, Italy.