  Given an open cover of a closed symplectic manifold, consider all smooth
partitions of unity consisting of functions supported in the covering sets. The
Poisson bracket invariant of the cover measures how much the functions from
such a partition of unity can become close to being Poisson commuting. We
introduce a new approach to this invariant, which enables us to prove the lower
bound conjectured by L. Polterovich, in dimension 2.
