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. 




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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.



Connectedness of moduli of vector bundles
Maybe the simplest example of the phenomenon mentioned by Jason is the case of rank 2 vector bundles on $\mathbb{P}^3$ with $c\_1$ even, studied in [this paper](http://gdz.sub.uni-goettingen.de/download/PPN356556735_0035/PPN356556735_0035___LOG_0016.pdf) of Atiyah and Rees. Besides $c\_1$ and $c\_2$ which encode the $K$-theory class, there is another topological invariant $\alpha $ with values in $\mathbb{Z}/2$, which can be $0$ or $1$ for holomorphic bundles with the same Chern classes.


