Let $X\subset \mathbb CP^n$ be a smooth submanifold whose normal bundle is 
$$\bigoplus\_{i=1}^{codim X}O(k\_i).$$


Is there some general enough additional condition of $X$ that implies that $X$ is a complete intersection?
For example, would $dimX\ge 2$ suffice (to exclude things like $X=\mathbb CP^1$)?



Projective submanifolds of $\mathbb CP^n$ whose normals bundles are sums of linear.  
I enter my comment as an answer. The paper *On the normal bundle of submanifolds of $\mathbb{P}^n$* by Lucian Badescu contains some answers to the question. Here are the links:


Published version: <http://www.ams.org/journals/proc/2008-136-05/S0002-9939-08-09255-1/>


On arXiv version: <http://arxiv.org/pdf/math/0701487v1.pdf>


In particular, Theorem 1.2 (due to Faltings) in the above reference is of interest, in connection to this question.


