Reference for triviality of deformation of holomorphic vector bundle Suppose $M$ and $X$ are complex manifolds with $X$ compact. Let $E\to M\times X$ be a holomorphic vector bundle. I am looking for a reference that contains a proof of the following statement:
If there exists $t\in M$ such that $H^1\left(X, \text{End}\left(E\right)\mid_{\left\{t\right\}\times X}\right)=0$, then there exists an open neighborhood $U\subset M$ of $t$ such that $E\mid_{\left\{t\right\}\times X}\cong E\mid_{\left\{s\right\}\times X}$ for all $s\in U$.
 A: One can think $E\to M\times X$ as a family of holomorphic vector bundles $E|_{\{t\}\times X}$ on $X$ parameterized by $t\in M$. It is mentioned in p. 198 of a paper  Deformations of Complex Structures and Holomorphic Vector Bundles by Narasimhan that the infinitesimal deformation of the vector bundle $E|_{\{t\}\times X}$ is given by a holomorphic map
$$T_xM\to H^1(X,End(E|_{\{t\}\times X})).$$
In particular it the group on the right-hand side vanishes, the local deformation is trivial, which is exactly what you want.
One may also wonder why it is $H^1(X,End(E|_{\{t\}\times X}))$ governs the infinitesimal deformation of a holomorphic vector bundle. There is a theory called differential graded Lie algebra (DGLA) guarantees that (see p.71 of Manetti's notes). In general, if $V$ is a holomorphic vector bundle on a complex manifold $X$, then $$L=\oplus_{p\ge 0}L^p=\oplus_{p\ge 0} \Gamma(X,\mathcal{A}^{0,p}(End(V)))$$
forms a DGLA, with $d=\bar{\partial}:L^p\to L^{p+1}$ satisfies $d^2=0$. So $L$ is a graded complex and its first cohomology $H^1(L)=H^1(X,End(V))$ is identified with the tangent space of the deformation space $\mathcal{Def}(L)$ of $L$, which is also the first order deformation space of holomorphic vector $V$ in our case.
