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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 $X=\mathbb CP^1$)?

edited Feb 9 at 15:03

Sasha
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asked Feb 9 at 14:54

aglearner
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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 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 at 21:06

Mahdi Majidi-Zolbanin · Accepted Answer · 2013-02-09 22:15:07Z UTC

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.

IMeasy · Answer 2 · 2013-02-09 15:42:31Z UTC

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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

answered Feb 9 at 15:42

IMeasy
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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 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 at 13:02

Projective submanifolds of $\mathbb CP^n$ whose normals bundles are sums of linear.

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