Title: On the structure of singularities of holomorphic flows in dimension 3

Authors: Julio C. Rebelo

Issue: Volume 59 (2010), Issue 3, 891-928

Abstract: 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.