  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.

Differential Geometry;Algebraic Geometry;Complex Variables