This question is related to, but not the same as, an earlier question: Koszul-Tate Resolution for Subvarieties of $\mathbb P^n$

Given a smooth projective variety $X\subset\mathbb P^n$, let $N_X$ be its normal bundle. Are there interesting examples of $X$ such that $N_X$ extends to a holomorphic vector bundle on all of $\mathbb P^n$?

If $X$ is a smooth complete intersection of hypersurfaces in $\mathbb P^n$, then $N_X$ obviously extends (so these examples are 'boring'). The only other examples I am aware of are the Horrocks-Mumford surfaces - see the answer by Sasha to my earlier question (linked in the beginning of this question).

  • $\begingroup$ Like Horrocks-Mumford bundle, I assume that a smooth variety coming from the zeroes of a section of a vector bundle of the right codimension would be considered boring. $\endgroup$ – Mohan Feb 6 at 3:05
  • $\begingroup$ Actually, I don't know any examples of smooth varieties cut out by a vector bundle of the right rank (other than complete intersections, the Horrocks-Mumford surfaces, and hypersurface sections of the Horrocks-Mumford surfaces). New examples of this kind would also be interesting to me. $\endgroup$ – Mohan Swaminathan Feb 6 at 3:59
  • $\begingroup$ On odd dimensional projective spaces, there are always non-split vector bundles of rank one less than the dimension, so you have curves in such spaces with the required property. $\endgroup$ – Mohan Feb 6 at 14:27
  • $\begingroup$ Could you please provide some more details or a reference where I could find a proof of your statement? $\endgroup$ – Mohan Swaminathan Feb 6 at 19:24

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