Cohomological and Poisson structures and integrable hierarchies in tautological subbundles for Birkhoff strata of the Sato Grassmannian

We consider cohomological and Poisson structures associated with the special tautological subbundles $TB_{W_{1,2,\dots,n}}$ for the Birkhoff strata of the Sato Grassmannian. We show that the tangent bundles of $TB_{W_{1,2, \dots,n}}$ are isomorphic to the linear spaces of two-coboundaries with vanishing Harrison cohomology modules. A special class of two-coboundaries is provided by a system of integrable quasilinear partial differential equations. For the big cell, it is the hierarchy of dispersionless Kadomtsev–Petvishvili (dKP) equations. We also demonstrate that the families of ideals for algebraic varieties in $TB_{W_{1,2,\dots,n}}$ can be viewed as Poisson ideals. This observation establishes a relation between families of algebraic curves in $TB_{W_{\widehat S}}$ and coisotropic deformations of such curves of zero and nonzero genus described by hierarchies of systems of hydrodynamic type; the dKP hierarchy is such a hierarchy. We note the interrelation between cohomological and Poisson structures.