# 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 $X=\mathbb CP^1$)?

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Perhaps this paper may answer your question: On the normal bundle of submanifolds of $\mathbb{P}^n$ by Lucian Badescu. Here is the link: arxiv.org/pdf/math/0701487.pdf –  Mahdi Majidi-Zolbanin Feb 9 '13 at 20:25
Mahdi, thank you for the link to the paper! This is exactly what I wanted :). Would you like to make this comment an answer, so that I could accept it? –  aglearner Feb 9 '13 at 21:06

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.

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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?"

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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

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IMeasy, a smooth submanifold of $\mathbb CP^n$ is always a local complete intersection, is not it? On the other hand the condition that I impose is clearly strong, so I am still optimstic about a possible positive answer to the question :) –  aglearner Feb 9 '13 at 18:19
Of course you are right. I am sorry I didn't notice that you assumed the smootheness of the variety. As it is now my comment means nothing. I will delete it in the next few days. ;) –  IMeasy Feb 10 '13 at 13:02