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The Runge approximation problem for holomorphic maps into Grassmannians. (English) Zbl 0827.32015
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.

32E30 Holomorphic, polynomial and rational approximation, and interpolation in several complex variables; Runge pairs
14M15 Grassmannians, Schubert varieties, flag manifolds
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