Manifold-valued holomorphic approximation (Q2882484)

Given a compact subset \(X\) of a complex space \(\mathcal X\) it is said that it admits holomorphic approximation if each continuous \(\mathbb C\)-valued function on it can be approximated uniformly on \( X\) by functions which are holomorphic on varying neighborhoods of \(X\) (in \(\mathcal X\)). The notion of \(\mathcal Y\)-valued holomorphic approximation, where \(\mathcal Y\) is another complex space is similar, yet \(\mathbb C\)-valued continuous functions have to be changed with \(\mathcal Y\)-valued continuous maps and holomorphic functions have to be changed with \(\mathcal Y\)-valued holomorphic maps. The author considers the following problem. Whether or not admitting holomorphic approximation is equivalent to admitting \(\mathcal Y\)-valued holomorphic approximation. For general complex spaces \(\mathcal Y\) this can not be true and simple counter-examples are to be found in this paper. The exact conditions ensuring equivalence are unknown, however, it turns out that if \(\mathcal Y\) is a complex manifold then \(\mathcal Y\)-valued holomorphic approximation follows from the holomorphic approximation property. The proof relies on reducing the general case to the easier and already studied case of a Stein manifold as target and this is achieved by using a Stein neighborhood of the totally real graph of a real embedding of \(\mathcal Y\) into some \(\mathbb R^p\).NEWLINENEWLINESimilar problems are considered also for holomorphic vector bundles.NEWLINENEWLINEIn the last two sections the rational approximation and restrictions on the dimension of \(X\) which admits holomorphic approximation are considered.NEWLINENEWLINEThe paper is motivated (and partially builds up on) earlier works by \textit{P. Gauthier} and \textit{E. S. Zeron} [Can. Math. Bull. 49, No. 2, 237--246 (2006; Zbl 1121.32006)] and \textit{E. S. Zeron} [Can. Math. Bull. 49, No. 4, 628--636 (2006; Zbl 1136.32005)].