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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the normal bundle of submanifolds of $ \mathbb{P}^n$


Author: Lucian Badescu
Journal: Proc. Amer. Math. Soc. 136 (2008), 1505-1513
MSC (2000): Primary 14M07, 14M10; Secondary 14F17
Published electronically: January 17, 2008
MathSciNet review: 2373577
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Abstract: Let $ X$ be a submanifold of dimension $ d\geq 2$ of the complex projective space $ \mathbb{P}^n$. We prove results of the following type.i) If $ X$ is irregular and $ n=2d$, then the normal bundle $ N_{X\vert\mathbb{P}^n}$ is indecomposable. ii) If $ X$ is irregular, $ d\geq 3$ and $ n=2d+1$, then $ N_{X\vert\mathbb{P}^n}$ is not the direct sum of two vector bundles of rank $ \geq 2$. iii) If $ d\geq 3$, $ n=2d-1$ and $ N_{X\vert\mathbb{P}^n}$ is decomposable, then the natural restriction map $ \mathrm{Pic}(\mathbb{P}^n)\to\mathrm{Pic}(X)$ is an isomorphism (and, in particular, if $ X=\mathbb{P}^{d-1}\times\mathbb{P}^1$ is embedded Segre in $ \mathbb{P}^{2d-1}$, then $ N_{X\vert\mathbb{P}^{2d-1}}$ is indecomposable). iv) Let $ n\leq 2d$ and $ d\geq 3$, and assume that $ N_{X\vert\mathbb{P}^n}$ is a direct sum of line bundles; if $ n=2d$ assume furthermore that $ X$ is simply connected and $ \mathscr O_X(1)$ is not divisible in $ \mathrm{Pic}(X)$. Then $ X$ is a complete intersection. These results follow from Theorem 2.1 below together with Le Potier's vanishing theorem. The last statement also uses a criterion of Faltings for complete intersection. In the case when $ n<2d$ this fact was proved by M. Schneider in 1990 in a completely different way.


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Additional Information

Lucian Badescu
Affiliation: Dipartimento di Matematica, Università degli Studi di Genova, Via Dodecaneso 35, 16146 Genova, Italy
Email: badescu@dima.unige.it

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09255-1
PII: S 0002-9939(08)09255-1
Keywords: Normal bundle, Le Potier's vanishing theorem, subvarieties of small codimension in the projective space.
Received by editor(s): June 19, 2006
Published electronically: January 17, 2008
Communicated by: Ted Chinburg
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.