Characterization on projective submanifolds of codimensions 2 and 3

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.