The Poisson bracket invariant on surfaces

We study the Poisson bracket invariant, which measures the level of Poisson noncommutativity of a smooth partition of unity, on closed symplectic surfaces. Motivated by a general conjecture of Polterovich [P3] and building on preliminary work of Buhovsky–Tanny [BT], we prove that for any smooth partition of unity subordinate to an open cover by discs of area at most c, and under some localization condition on the cover when the surface is a sphere, then the product of the Poisson bracket invariant with c is bounded from below by a universal constant. Similar results were obtained recently by Buhovsky–Logunov–Tanny [BLT] for open covers consisting of displaceable sets on all closed surfaces, and their approach was extended by Shi–Lu [SL] to open covers by nondisplaceable discs. We investigate the sharpness of all these results.