Let \(X\) be a complex algebraic manifold, \(K\) a compact holomorphically convex subset of \(X\), and \(Y\) an algebraic manifold homogeneous for some complex linear algebraic group. The main theorem states that a holomorphic map \(f\colon K\to Y\) can be approximated by regular maps from \(K\) to \(Y\) if and only if \(f\) is homotopic to a regular map from \(K\) to \(Y\). Consequently, every null homotopic holomorphic map from \(K\) to \(Y\) can be approximated by regular maps from \(K\) to \(Y\). The authors provide an example of a null homotopic holomorphic map from \(K\) to \(Y\) that does not admit approximation by regular maps from \(X\) to \(Y\). It is known that if \(Y\) is algebraically subelliptic, then every holomorphic map from \(K\) to \(Y\) can be approximated uniformly on \(K\) by regular maps from \(X\) to \(Y\).
The authors define cascades, which is inspired by M. Gromov’s notion of spray [J. Am. Math. Soc. 2, No. 4, 851–897 (1989; Zbl 0686.32012)], and the proofs follow closely [F. Forstnerič, Am. J. Math. 128, No. 1, 239–270 (2006; Zbl 1171.32303)] for algebraically subelliptic manifolds. The paper contains many nontrivial examples, among them those showing that algebraic homogeneity and algebraic subellipticity are complementary properties: neither implies the other.