On the structure of singularities of holomorphic flows in dimension 3 Julio Rebelo 32S65vector fieldssingular foliations We study singularities of semi-complete holomorphic vector fields in dimension $3$. Under the very generic assumption that the singularities of the associated foliation can be resolved with a single blow-up, we establish Ghys's conjecture asserting that the second jet of the initial vector field cannot vanish at an isolated singularity. In fact, it results from our proof that, under the preceding assumption, it suffices to assume that the singular set of the mentioned vector field has codimension at least $2$. Related to this question and, in fact, playing a central role in our proof, we also present a detailed picture of the semi-complete singularities of vector fields whose associated foliation has a single eigenvalue different from zero. Indiana University Mathematics Journal 2010 text pdf 10.1512/iumj.2010.59.3980 10.1512/iumj.2010.59.3980 en Indiana Univ. Math. J. 59 (2010) 891 - 928 state-of-the-art mathematics http://iumj.org/access/