Geometry of operator cross ratio

The operator cross ratio, which is meaningful, in particular, for the infinite-dimensional Sato Grassmannian is defined and investigated. Its homological interpretation is presented. A matrix and operator analogue of the Schwartzian differential operator is introduced and its relation to linear Hamiltonian systems and Riccati's equation is established. The aim of these constructions is application to the KP-hierarchy (the Kadomtsev-Petviashvili hierarchy).