Feigin-Odesskii brackets associated with Kodaira cycles and positroid varieties

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 $C^n$, using the shifted Poisson structure on the latter moduli spaces and relating them to a certain twist of the standard Poisson structure on $G(k,n)$. %by a bivector field on its maximal torus. This link allows us to solve a classification problem for extensions of vector bundles over $C^n$. Based on this solution we further classify the symplectic leaves of all positroid varieties in $G(k,n)$ with respect to the twisted standard Poisson structure. Moreover, we get an explicit description of the moduli stack of symplectic leaves of $G(k,n)$ with the twisted standard Poisson structure as an open substack of the stack of vector bundles on $C^n$.