With Sasha Polishchuk we construct a 0-shifted Poisson structure on moduli stack of bounded complexes of vector bundles on Gorenstein CY curves modulo chain isomorphisms. This Poisson structure “derives” many classical Poisson structures that appear in algebraic geometry, representation theory and integrable systems.
Using the derived Lagrangian intersection, we determine the derived symplectic leaves of this Poisson structure. In this talk I will present an example when the curve is a Kodaira cycle. On one component of the moduli stack our Poisson structure specialises in Drinfeld's standard Poisson structure on Grassmannians. In particular we obtain a classification of symplectic leaves of the standard Poisson structure. This example is closely related to Lusztig’s total positivity where the torus orbits of symplectic leaves are the Lusztig strata.
