Characterization on projective submanifolds of codimensions 2 and 3

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.

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