  Cohomological and Poisson structures associated with the special tautological
subbundles $TB_{W_{1,2,\dots,n}}$ for the Birkhoff strata of Sato Grassmannian
are considered. It is shown that the tangent bundles of $TB_{W_{1,2,\dots,n}}$
are isomorphic to the linear spaces of $2-$coboundaries with vanishing
Harrison's cohomology modules. Special class of 2-coboundaries is provided by
the systems of integrable quasilinear PDEs. For the big cell it is the dKP
hierarchy. It is demonstrated also that the families of ideals for algebraic
varieties in $TB_{W_{1,2,\dots,n}}$ can be viewed as the Poisson ideals. This
observation establishes a connection between families of algebraic curves in
$TB_{W_{\hat{S}}}$ and coisotropic deformations of such curves of zero and
nonzero genus described by hierarchies of hydrodynamical type systems like dKP
hierarchy. Interrelation between cohomological and Poisson structures is noted.
