Geometry of cross ratio

Abstract

Generalization of the cross ratio to polarizations of linear finite and infinite-dimensional spaces (in particular to Sato Grassmannian) is given and explored. This cross ratio appears to be a cocycle of the canonical (tautalogical) bundle over the Grassmannian with coefficients in the sheaf of its endomorphisms. Operator analog of the Schwarz differential is defined. Its connections to linear Hamiltonian systems and Riccati equations are established. These constructions aim to obtain applications to KP-hierarchy.