Universal property of chern character forms of the canonical connection

Let _q,n denote the complex Stiefel bundle over the complex Grassmannian and let ₀ be the universal connection on this bundle. Consider the Chern character form of ₀ defined by the formula where ₀ is the curvature form of the connection ₀. Let M be a manifold of dimension m and a closed 2k-form on M. Suppose, there exists a continuous map which pulls back the cohomology class of ch_k(₀) onto the cohomology class of . We prove that if q and n are greater than certain numbers (which we determine in this paper) then there exists a smooth map such that f*ch_k(₀) = .