Let $X$ be a smooth projective variety. Given two vector bundles $V_1$ and $V_2$ such that $[V_1]=[V_2]\in K^0(X)$, can one expect that $V_1$ and $V_2$ can be connected by a family of vector bundles? Or are there any counterexamples?
For example, if we have a short exact sequence of vector bundles
$$0\rightarrow E_0\rightarrow E_1\rightarrow E_2\rightarrow 0 $$,
then $E_1$ can be deformed to $E_0\oplus E_2$ by finding a curve between corresponding elements in $Ext^1(E_2,E_0)$. Since the relations in $K^0$ are generated by short exact sequences of vector bundles, I was attempting to generalize this kind of argument but failed.
Another motivation for this question is that I want to know the possibility of identifying the Gromov-Witten theories of $\mathbb P(V_1)$ and $\mathbb P(V_2)$ for such $V_1$ and $V_2$ by deforming one to another. Therefore I think I would like to see, for example, families of vector bundles over a reducible variety.