Let $C$ be a Riemann Surface of genus $g \geq 2$. Consider a Vector Bundle of rank $r$ and degree $d$ on $C$. It is often convenient to construct such a Vector Bundle as an extension
\begin{equation} 0 \rightarrow E' \rightarrow V \rightarrow E'' \rightarrow 0 \end{equation}
where $E'$ and $E''$ are "smaller" bundles.
Now, I believe it is in general quite a non-trivial problem to study whether two distinct extensions can lead to bundles that are isomorphic. A simple instance of this is a case when the space of extensions is one dimensional. That is, $Ext^1(E'',E') = \mathbb{C}$. For any non-zero element of $Ext^1(E'',E')$, one gets a distinct (non-trivial) extension, but the resulting bundles are isomorphic. On the other hand, the zero element $0 \in Ext^1(E'',E') $ corresponds to the split extention. The corresponding bundle is just the direct sum $E' \oplus E''$.
My post here concerns cases when the space $Ext^1(E'',E')$ is bigger.
First, how much is known about when such extensions lead to non-isomorphic bundles ?
Is Question 1. particularly easier to answer for (semi-)stable bundles vs unstable bundles ? (I am thinking here of Mumford's notion of stability for vector bundles)
If Answer to 2. is yes, why is that the case ? I am really hoping that the question can be (or has been) answered for atleast for some classes of unstable bundles.
- I would also like to know if this sort of question has been discussed in the context of $G$ principal bundles for $G \neq GL_n$
One instance where I have seen a non-trivial example discussed is in Newstead's book "Lectures on Introduction to Moduli Problems and Orbit Spaces". In Chapter 5 of this book, he recalls an example due to Atiyah which concerns rank 2 bundles in the g = 2 case. He uses this example to show "Jump Phenomenon" for non-decomposable bundles and notes that the general problem appears to be quite difficult. In the years since, have more such examples been worked out ?
The Atiyah-Newstead example also suggests that the stable bundles case may be a very convenient setting to answer this sort of question. But, I am really hoping the question can be answered for a larger class of bundles.
