Universal property of Chern character forms of the canonical connection (Q1774011)

For the complex Grassmannian \(GR_n(\mathbb{C}^q)\) there is a closed \(2k\)-form defining the Chern character \(ch_k(\omega_0)\). This paper proves a universality property of this form. If \(M\) is a manifold of dimension at most \(m\) with a closed \(2k\)-form \(\sigma\) for which there is a continuous map \(f_0: M \rightarrow GR_n(\mathbb{C}^q)\) with \(n,q\) sufficiently large for which the cohomology class of \(ch_k(\omega_0)\) pulls back to that of \(\sigma\), then there is a smooth map \(f: M \rightarrow GR_n(\mathbb{C}^q)\) for which \(f^*ch_k(\omega_0) = \sigma\).