Feigin-Odesskii brackets associated with Kodaira cycles and positroid varieties

Abstract: We establish a link between open positroid varieties in the Grassmannians $G(k,n)$ and certain moduli spaces of complexes of vector bundles over Kodaira cycle $C<sup>n$, using the shifted Poisson structure on the latter moduli spaces and relating them to the standard Poisson structure on $G(k,n)$. This link allows us to solve a classification problem for extensions of vector bundles over $C<sup>n$. Based on this solution we further classify the symplectic leaves of all positroid varieties in $G(k,n)$ with respect to the standard Poisson structure. Moreover, we get an explicit description of the moduli stack of symplectic leaves of $G(k,n)$ with the standard Poisson structure as an open substack of the stack of vector bundles on $C<sup>n$.