The Euler and Pontrjagin numbers of an n-manifold in \({\mathbb{C}}^ n\) (Q1059189)

A compact real n-manifold M immersed or embedded in \({\mathbb{C}}^ n\) or other complex n-manifold is considered. The main emphasis is on the relation between the configuration of complex tangents to M and the characteristic numbers of M. One result is a formula for the Euler number of M, under suitable restrictions on the immersion. This generalizes earlier results of E. Bishop and R. Wells. A formula for the Pontrjagin number of a 4-manifold M generically immersed in \({\mathbb{C}}^ 4\) is also given. It has the consequence that, for \(M={\mathbb{P}}_ 2{\mathbb{C}}\), there is a non-trivial local hull of holomorphy. This second formula has subsequently been extended to the context of 4-plane bundles over a 4- manifold.