Let $X$ be a smooth projective variety and let $Y\subset X$ be a smooth subvariety. Given a vector bundle $E$ on $Y$, when can $E$ be extended to a vector bundle $\tilde E$ on $X$? I.e., are there cohomology groups containing the obstructions to extending $E$? 



Extending vector bundles from subvarieties
Without hypotheses on $Y$ there is no hope to define such obstructions, already for line bundles. A natural hypothesis is to take for $Y$ a (smooth) ample divisor in $X$, of dimension $\geq 2$. In [this paper](http://projecteuclid.org/download/pdf_1/euclid.jmsj/1239802165), Fujita gives some cohomological conditions which imply that $E$ extends : $H^2(Y, \mathcal{E}nd(E)(-tY))=0$ for all $t\geq 1$ and $H^p(Y, E(tY))=0$ for all $t\in\mathbb{Z}$ and $0<p<\dim Y$. Be aware that these conditions are extremely strong.


