I am not sure how much obvious or wrong is the following question:

For every (holomorphic) vector bundle over a complex projective variety, is it possible to deform it to another one which is decomposable (into line bundles)? I feel the answer is yes (since obstruction lies in the ext group and you can deform any element of ext group to zero) but I don't know how easy is a precise proof!

Is this true at least over curves?

After edit:

How do we show that for every (holomorphic) vector bundle over a curve, is it possible to deform it to another one which is decomposable (into line bundles)?

Before edit:

I am not sure how much obvious or wrong is the following question:

For every (holomorphic) vector bundle over a complex projective variety, is it possible to deform it to another one which is decomposable (into line bundles)? I feel the answer is yes (since obstruction lies in the ext group and you can deform any element of ext group to zero) but I don't know how easy is a precise proof!

Is this true at least over curves?

I am not sure how much obvious or wrong is the following question:

For every (holomorphic) vector bundle over a complex projective variety, is it possible to deform it to another one which is decomposable (into line bundles)? I feel the answer is yes (since obstruction lies in the ext group and you can deform any element of ext group to zero) but I don't know how easy is a precise proof!

I am not sure how much obvious or wrong is the following question:

For every (holomorphic) vector bundle over a complex projective variety, is it possible to deform it to another one which is decomposable (into line bundles)? I feel the answer is yes (since obstruction lies in the ext group and you can deform any element of ext group to zero) but I don't know how easy is a precise proof!

Is this true at least over curves?