  We study germs of singular holomorphic vector fields at the origin of $\Bbb
C^n$ of which the linear part is 1-resonant and which have a polynomial normal
form. The formal normalizing diffeomorphism is usually divergent at the origin
but there exists holomorphic diffeomorphisms in some "sectorial domains" which
transform these vector fields into their normal form. In this article, we study
the interplay between the small divisors phenomenon and the Gevrey character of
the sectorial normalizing diffeomorphisms. We show that the Gevrey ordrer of
the latter is linked to the diophantine type of the small divisors.
