Necessary and sufficient conditions for meromorphic integrability near a curve

Abstract

Let us consider a vector field $X$ meromorphic on a neighbourhood of an algebraic curve $\bar{\Gamma}\subset \mathbb{P}^n$ such that $\Gamma$ is a particular solution of $X$. The vector field $X$ is $(l,n-l)$ integrable if it there exists $Y_1,\dots,Y_{l-1},X$ vector fields commuting pairwise, and $F_1,\dots,F_{n-l}$ common first integrals. The Ayoul-Zung Theorem gives necessary conditions in terms of Galois groups for meromorphic integrability of $X$ in a neighbourhood of $\Gamma$. Conversely, if these conditions are satisfied, we prove that if the first normal variational equation $NVE_1$ has a virtually diagonal monodromy group $Mon(NVE_1)$ with non resonance and Diophantine properties, $X$ is meromorphically integrable on a finite covering of a neighbourhood of $\Gamma$. We then prove the same relaxing the non resonance condition but adding an additional Galoisian condition, which in fine is implied by the previous non resonance hypothesis. Using the same strategy, we then prove a linearisability result near $0$ for a time dependant vector field $X$ with $X(0)=0\;\forall t$.