The Hartogs extension theorem for holomorphic vector bundles and sprays

We give a detailed proof of Siu’s theorem on extendibility of holomorphic vector bundles of rank larger than one, and prove a corresponding extension theorem for holomorphic sprays. We apply this result to study ellipticity properties of complements of compact subsets in Stein manifolds. In particular we show that the complement of a closed ball in C n , n ≥ 3 , is not subelliptic.