# Vector bundles trivial on complement of a hypersurface section

Assume we have a smooth projective variety $$X$$ over a field (finite field if matters). Consider a smooth hypersurface section $$H$$ of $$X$$. Let $$Spec(A)=U=X \setminus H$$ be an affine variety. I want to know whether there are any nice description of vector bundles on $$X$$ such that they are trivial on $$U$$ and $$H$$? My suspicion was that they are all direct sums of line bundles of the form $$O(nH)$$ is that correct?

Edit: Let's assume that we are talking about rank $$m$$ vector bundles that are subsheaves of the constant sheaf given by the vector space $$F(X)^{\oplus m}$$ which $$F(X)$$ is the function field or equivalently field of fractions of $$A$$. Does that change anything?

• This is not true. Try the projective plane and $H$ a hyeprplane. – Mohan Apr 10 '19 at 1:33
• @Mohan Is this true if we assume the variety is 1 dimensional? And assume the $H$ is just 1 point. – user127776 Apr 10 '19 at 3:29
• This is definitely false. Take a smooth projective curve $X$ of genus $\geq 2$. There are of course many nontrivial vector bundles on $X$ with trivial determinant. Their restriction to the complement of a point is trivial, because a projective module over a Dedekind ring with trivial determinant is free. – abx Apr 10 '19 at 5:24
• I made an edit, I'd really appreciate your opinions on it. – user127776 Apr 10 '19 at 6:55
• All rank $m$ vector bundles embed into $F(X)^{\oplus m}$ (by restriction to the generic fiber). – abx Apr 10 '19 at 7:07

In many cases, $$\mathcal{O}_X(n \cdot H)$$ itself will not be trivial on both $$H$$ and $$U$$, because it will be nontrivial on $$H$$. For example, if $$H \subset \mathbb{P}^n$$ is a degree $$d$$ hypersurface, then $$\mathcal{O}_{\mathbb{P}^n}(H) \simeq \mathcal{O}_{\mathbb{P}^n}(d)$$ and so $$\mathcal{O}_{\mathbb{P}^n}(H)|_H \simeq \mathcal{O}_H(d)$$ which is non-trivial as long as $$n> 1$$, for instance because it is ample.
EDIT: I realize what this shows is: not all sums of line bundles of the form $$\cal{O}_X(n \cdot H)$$ are trivial on both $$U$$ and $$H$$. All the same it seemed relevant.