Constructions of Poisson structures on smooth 4-manifolds and a potential invariant

In joint work with Garcia-Naranjo and Vera, we adapted rank-2 Poisson structures to broken Lefschetz fibrations. We provided local descriptions of the Poisson bivectors and the symplectic forms of its symplectic foliation. I want to discuss using  a unique property of these Poisson structures to define an invariant. We can find a specific surface for each Poisson structure (coming from a broken or wrinkled Lefschetz fibration). It lives in a class that is dual to the contraction of an unimodular volume form with respect to the Poisson bivector. How small could this surface be?