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

TL;DR

This paper explores the cohomological and Poisson structures of tautological subbundles in the Birkhoff strata of the Sato Grassmannian, linking integrable PDEs, algebraic geometry, and deformation theory.

Contribution

It establishes isomorphisms between tangent bundles and 2-coboundaries, connects algebraic varieties with Poisson ideals, and relates cohomological and Poisson structures in integrable hierarchies.

Findings

Tangent bundles are isomorphic to spaces of 2-coboundaries.

Families of algebraic variety ideals are Poisson ideals.

Connections between cohomological and Poisson structures are demonstrated.

Abstract

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.