I had a number of questions that are somewhat related to each other. I decided to post them altogether instead of separately. I'd appreciate any kinds of answers, ideas or sources regarding any of them. Throughout these questions $X$ is a smooth variety and $Z$ is a closed smooth subvariety (you can assume it is of codimension 1). The complement is denoted by $U$.
Q1) This is my main question which is a special case of Q4 but I bring it first because of its importance to me. Here assume $X=Y\times \mathbb{A}^1$ where $Y$ is a smooth projective variety over a field. $Z=Y\times \{0\}$. Given two vector bundles on $X$ where they are isomorphic on $U$ and $Z$, is it possible to deform one of them to other, where throughout the deformation the restriction to $U$ and $Z$ remain the same vector bundles? (In other words find a vector bundle on $X\times \mathbb{A}^1$ where its restrictions at times $0$ and $1$ gives us the original two vector bundles and the restriction to $U\times \mathbb{A}^1$ and $Z\times \mathbb{A}^1$ are just extended vector bundles from $U$ and $Z$ respectively.). Note that here the obstruction in $K_0$ or chern classes vanishes automatically.
Q2) Let's denote the formal completion of $X$ along $Z$ by $\hat{X}_Z$. If $E$ is a vector bundle on $Z$ is there any nice obstructions for the extension of $E$ to $\hat{X}_Z$? If so and this obstruction vanishes is there any nice description of all these lifts? (For example: Do they form certain vector space or not?) If we replace the formal completion with order 1 thickening there is such a description where the obstruction lives in $H^2(Z,I\otimes \mathcal{E}nd(E))$ where $I$ is the ideal corresponding to $Z$. You can find more details here page 34 Theorem 5.3.
Q3) Given two extensions of $E$ to $\hat{X}_Z$, denoted by $E_1$ and $E_2$, is it possible to deform $E_1$ to $E_2$ while the restriction to $Z$ should be $E$ throughout the deformation? This means finding a vector bundle on $\hat{X}_Z\times \mathbb{A}^1$ such that it is the extension of $E$ on $Z\times \mathbb{A}^1$ and gives us vector bundles $E_1$ and $E_2$ at times $0$ and $1$. I'd imagine something like this should follow by some sort of affineness of all extensions of $E$ given the obstruction is zero(Basically Q1). Somewhat similar to deforming any two extensions to each other, in the $Ext$ group.
Q4) Given two vector bundles $V_1$ and $V_2$ on $X$ such that $V_1|_{Z}\cong V_2|_{Z} \cong E$ and $V_1|_{U}\cong V_2|_U \cong F$. Is it possible to deform $V_1$ to $V_2$ while throughout the deformation the restriction to $Z$ should be isomorphic to $E$ and the restriction to $U$ should be isomorphic to $F$? (With the similar definition of deformation given in Q2). An obvious obstruction is $K_0$ so let's assume $V_1$ and $V_2$ have the same image in $K_0(X)$
There is a descent that is given in this paper, basically claims any coherent sheaf can be uniquely determined by its restriction to $U$, the tubular neighborhood $\hat{X}_Z$ and some sort of patching data in the punctured tubular neighborhood. The punctured neighborhood is defined using Berkovich spaces which I'm not too familiar with them. But I think it should boil down to Q2 and deforming in the punctured nbhd (if true).
