Ôike, Hiroshi Non-existence of higher order non-singular holomorphic immersions. (English) Zbl 0577.32028 Hokkaido Math. J. 13, 251-259 (1984). The purpose of this note is to study holomorphic immersions of the form \(f: {\mathbb{P}}_ n\to {\mathbb{P}}_ N\) or more generally \(f: V\to {\mathbb{P}}_ N\) where \({\mathbb{P}}_ n\) and \({\mathbb{P}}_ N\) are complex projective spaces and V is a nonsingular hypersurface in \({\mathbb{P}}_{n+1}\). The main results are inequalities relating n, N, the degree of f, the degree of V and the least order of an inflection point of f. The proof uses the symmetric power operators in K-theory. The inequalities are obtained by studying the Chern classes of holomorphic vector bundles E associated to the map f and observing that if \(c_ m(E)\) is nonzero then the rank of E is at least m. Reviewer: F.Kirwan MSC: 32J25 Transcendental methods of algebraic geometry (complex-analytic aspects) 32C15 Complex spaces 32H99 Holomorphic mappings and correspondences 14C30 Transcendental methods, Hodge theory (algebro-geometric aspects) 55R50 Stable classes of vector space bundles in algebraic topology and relations to \(K\)-theory Keywords:non-existence of holomorphic immersions; complex projective spaces; inflection point PDFBibTeX XMLCite \textit{H. Ôike}, Hokkaido Math. J. 13, 251--259 (1984; Zbl 0577.32028) Full Text: DOI