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.

中文翻译：

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曲面上统一分区的泊松括号

给定一个封闭辛流形的开覆盖，考虑所有由覆盖集支持的函数组成的统一的光滑分区。覆盖的泊松括号不变量测量来自这种统一划分的函数可以变得接近泊松交换的程度。我们为这个不变量引入了一种新方法，它使我们能够证明 L. Polterovich 在维度 2 中推测的下界。