MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 $X=\mathbb CP^1$)?

share|cite|improve this question
Perhaps this paper may answer your question: On the normal bundle of submanifolds of $\mathbb{P}^n$ by Lucian Badescu. Here is the link: – 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
up vote 4 down vote accepted

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:

On arXiv version:

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

share|cite|improve this answer

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?"

share|cite|improve this answer

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

share|cite|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.