Universal property of chern character forms of the canonical connection

Abstract.

Let γ_q,n denote the complex Stiefel bundle over the complex Grassmannian $Gr_n (\mathbb{C}^q )$ and let ω₀ be the universal connection on this bundle. Consider the Chern character form of ω₀ defined by the formula $ch(\omega _0 ) = \sum\nolimits_{k \geq 0} {\frac{1} {{k!}}ch_k } (\omega _0 ) = \sum\nolimits_{k \geq 0} {\frac{1} {{k!}}{\text{trace}}(i\Omega _0 )^k ,}$ where Ω₀ is the curvature form of the connection ω₀. Let M be a manifold of dimension ≤ m and σ a closed 2k-form on M. Suppose, there exists a continuous map $f_0 :M \to Gr_n (\mathbb{C}^q )$ which pulls back the cohomology class of ch _k (ω₀) onto the cohomology class of σ. We prove that if q and n are greater than certain numbers (which we determine in this paper) then there exists a smooth map $f : M \to Gr_n (\mathbb{C}^q )$ such that f*ch _k (ω₀) = σ.