Decomposability of a Poisson Tensor Could Be a Stable Phenomenon

Abstract

In this paper, we develop one of the questions raised by the author in the mini-course he gave at the conference Geometry and Physics V held at the University Cheikh Anta Diop, Dakar in May 2007). Let $\Pi$ be a Poisson tensor on a manifold $M.$ We suppose that it is decomposable in a neighborhood $U$ of a point $m,$ i.e. we have $\Pi=X\wedge Y$ on $U$ where $X$ and $Y$ are two vector fields. We will exhibit examples where every Poisson tensor near enough $\Pi$ seems to be also decomposable in a neighborhood of a point which can be chosen arbitrarily near $m$; and this works even if $M$ has a big dimension. This idea is a consequence of a cohomology calculation which can be interesting by itself.