Characterization on projective submanifolds of codimensions 2 and 3

TL;DR

This paper provides a complete characterization of projective submanifolds with codimensions 2 and 3 in projective space, based on Chern classes and line bundles, extending previous results for hypersurfaces.

Contribution

It offers necessary and sufficient conditions for such submanifolds, generalizing earlier work on hypersurfaces to higher codimensions.

Findings

Characterization conditions involve Chern classes and very ample line bundles.

Extends previous hypersurface characterization to codimensions 2 and 3.

Proposes higher codimensional cases as an open question.

Abstract

In this article we give a necessary and sufficient condition to characterize projective submanifolds in ${\mathbb P}^N$ with codimensions 2 and 3. The conditions involve the Chern classes of the manifold and a very ample line bundle on the manifold. This generalizes our earlier characterization for hypersurfaces. The higher codimensional cases are proposed as a general question.