Feigin–Odesskii brackets associated with Kodaira cycles and positroid varieties

Abstract

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