Approximation of holomorphic mappings on 1-convex domains

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Let π:Z→ X be a holomorphic submersion of a complex manifold Z onto a complex manifold X and D X a 1-convex domain with strongly pseudoconvex boundary. We prove that under certain conditions there always exists a spray of π-sections over D which has prescribed core, it fixes the exceptional set E of D, and is dominating on D E. Each section in this spray is of class Cᵏ(D) and holomorphic on D. As a consequence we obtain several approximation results for π-sections. In particular, we prove that π-sections which are of class Cᵏ(D) and holomorphic on D can be approximated in the Cᵏ(D) topology by π-sections that are holomorphic in open neighborhoods of D. Under additional assumptions on the submersion we also get approximation by global holomorphic π-sections and the Oka principle over 1-convex manifolds. We include an application to the construction of proper holomorphic maps of 1-convex domains into q-convex manifolds.