holomorphic tubular neiborhood and meromorphic vector fields

par Hamza Bakhouch

vendredi 3 juil. 2026, 10:30 → 12:30 Europe/Paris

It is known that the existence of a holomorphic tubular neighborhood of a compact submanifold of a compact complex manifold is rare; for example, the classical theorem due to Van de Ven states that the only submanifolds of CP(n) that possess a holomorphic tubular neighborhood are the linear ones. However, when X is a meromorphic vector field in a neighborhood of a submanifold S such that the associated foliation leaves S invariant, we show—by choosing the right coordinates (admissible coordinates)—that X induces a meromorphic vector field X_{00} on the whole of the normal bundle of S, and the new foliation leaves S invariant as well. Moreover, our motivation for this construction comes from the study of polynomial complete vector fields on \mathbb{C}^n, so the main focus is to understand the induced foliation on the hyperplane at infinity (S), which happens to be equivalent to study the restriction of the foliation induced by X_{00} to S. This talk is devoted to give an idea about the construction, as well as how it preserves the character of the vector field, mainly the semicompleteness, and how this construction will allow us to use some algebraic geometry tools for work in progress.