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).