I am studying a certain problem arising in holomorphic vector bundles over a complex manifold $X$.
My goal is to understand how to catalogue the possible extensions of two given (holomorphic) vector bundles $E_1,E_2$ over $X$, i.e. all possible $E$ fitting a exact sequence of the form $$ 0 \rightarrow E_1 \rightarrow E \rightarrow E_2 \rightarrow 0$$ I have been directed towards something denoted as $H^1(X,E_1\otimes E_2^*)$, which I suppose is certain cohomology group, as a parametrization for these extensions $E$.
I would like to learn precisely how to construct such an extension from this group explicitly.
As reference, my background encompasses de Rham cohomology and the notion of sheaves, besides the bijection between locally free $\mathcal{O}_X$-modules and holomorphic vector bundles, and a little bit of Dolbeault cohomology, but I lack many of the basics of algebraic geometry.
Any outline of where to even start reading would be appreciated.
This is not necessairly the ideal answer, but i would say the following.
The vector bundle $E_2^* \otimes E_1$ has the local endomorphisms from $E_2$ to $E_1$ as local sections.
Over an open $U$ where the two vector bundles are trivial, the exact sequence splits so there is a $s_U : E_2|_U \longrightarrow E|_U$ such that $p|_U \circ s_U = Id$ where $p$ stands for the second arrow.
Over an intersection, when you compose $p$ with the difference $s_{U_1}-s_{U_2}$ it must therefore be zero : so this difference has values in $E_1 = ker(p)$.
So $(s_{U_1}-s_{U_2})_{U_1,U_2}$ is a cocycle with values in $E_2^* \otimes E_1$.
Conversely, you can glue together the directs sums $E_1|_U \oplus E_2|_U$ using the transitions $\psi_{1,2},\varphi_{1,2}$ of $E_1$ and $E_2$ and a cocycle as previously. In an explicit way, you have the sum of two trivial bundles over $U_1$ and $U_2$ and the transition is a matrice $$\begin{pmatrix} \psi_{1,2} &s_{1,2}\\ 0 & \varphi_{1,2} \end{pmatrix}$$ where $s_{1,2}$ is the matrice of the endomorphism corredponding to the cocycle over $U_1 \cap U_2$.