Homotopy BV algebras in Poisson geometry

We define and study the degeneration property for $\mathrm{BV}_\infty$ algebras and show that it implies that the underlying $L_\infty$ algebras are homotopy abelian. The proof is based on a generalisation of the well- known identity $\Delta(e^\xi)=e^\xi\left(\Delta(\xi)+\frac12[\xi,\xi]\right)$ which holds in all BV algebras. As an application we show that the higher Koszul brackets on the cohomology of a manifold supplied with a generalised Poisson structure all vanish. References: 17 entries.