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Non-existence of higher order non-singular holomorphic immersions. (English) Zbl 0577.32028

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
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