The Runge approximation problem for holomorphic maps into Grassmannians
Let \(X\) be a complex affine algebraic variety and let \(f : X \to \mathbb{G}_{n, p}\) be a holomorphic map into the Grassmannian of \(p\)- dimensional complex vector subspaces of \(\mathbb{C}^n\). Assume that \(p + \dim X \leq n\). It is proved that \(f\) can be approximated by regular maps (that is, algebraic morphisms) if and only if the pull-back vector bundle \(f^* \gamma_{n,p}\), where \(\gamma_{n,p}\) is the universal vector bundle on \(\mathbb{G}_{n,p}\), admits an algebraic structure. Some applications are also discussed.
scientific article; zbMATH DE number 778949