# Extending Normal Bundle of a Subvariety of $\mathbb P^n$

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).

• 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. – Mohan Feb 6 at 3:05
• 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. – Mohan Swaminathan Feb 6 at 3:59
• 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. – Mohan Feb 6 at 14:27
• Could you please provide some more details or a reference where I could find a proof of your statement? – Mohan Swaminathan Feb 6 at 19:24