Projective submanifolds of $\mathbb CP^n$ whose normals bundles are sums of linear.   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 $X=\mathbb CP^1$)?
 A: 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.
A: For sure if the variety is a complete intersection then the normal bundle splits. I am afraid that the other way round is not true. IMO it should imply just being locally complete intersection, and a priori there's no general condition that implies that a l.c.i. is a c.i., as far as I know
A: If $X\subset\mathbb{P}^n$ is a smooth subvariety and the normal bundle splits then, by adjunction, $X$ is subcanonical i.e. $\omega_{X}\cong\mathcal{O}_X(k)$. This is important for the following result in codimension two.
In "Bénédicte Basili and Christian Peskine, Décomposition du fibré normal des surfaces lisses de $\mathbb{P}^4$ et structures doubles sur les solides de $\mathbb{P}^5$, Duke Math. J.
Volume 69, Number 1 (1993), 1-245", you can find the following result:
Let $X\subset\mathbb{P}^N$, $n\geq 4$ be a smooth codimension two subvariety. If $N_X$ splits then $X$ is a complete intersection. 
For curves in $\mathbb{P}^3$ is quite different and the following is still an open problem: "Let $C\subset\mathbb{P}^3$ be a smooth, connected curve. Is it true that if $N_C = \mathcal{O}_C(a)\oplus\mathcal{O}_C(b)$, then $C$ is a complete intersection?"
