Let $X$ be a compact complex manifold and $E$ be a finite dimensional holomorphic vector bundle on $X$ with a fixed $\bar{ \partial}$-connection $\bar{\partial}_E$.
Now we consider a small neighborhood $\Delta$ of $0$ in $\mathbb{C}$. Let $\epsilon_1(t)$ and $\epsilon_2(t)$ be two deformations of $\bar{\partial}_E$, i.e. they satisfy
- $(\bar{\partial}_E+\epsilon_i(t))^2=0$, $i=1,2$.
- $\epsilon_i(t)$ is holomorphic with respect to $t$, i.e. $\epsilon_i(t)\in \Omega^{0,1}(X, \text{End}(E))\otimes \mathcal{O}(\Delta)$, $i=1,2$.
- $\epsilon_i(0)=0$, $i=1,2$.
Let $\phi(t)\in C^{\infty}(X\times \Delta,\text{End}(E))$ be such that
- $\phi(t)\circ (\bar{\partial}_E+\epsilon_1(t)=(\bar{\partial}_E+\epsilon_2(t))\circ \phi(t)$.
- $\phi(0)\equiv \text{id}_E$ on $X$.
Then do we always have $\phi(t)$ is holomorphic with respect to $t$? If not, do we have counter examples?
Edit: The answer to the original question is trivially false: For example let $E=\mathbb{C}$ be the trivial line bundle with $\bar{\partial}_E=\bar{\partial}_X$. Let $\epsilon_1(t)=\epsilon_2(t)\equiv 0$. Let $\phi(t)=1+f(t)$ where $f(t)$ is a $C^{\infty}$, non-holomorphic function of $t$ such that $f(0)=0$, considered as a constant function on $X$ for each $t$. Then $\phi(t)$ gives a counter example.
Therefore I think the right question to ask is
If there exists a $\phi(t)$ satisfying the above Condition 1 and 2, could we find a $\psi(t)$ which also satisfies Condition 1 and 2 and is holomorphic with respect to $t$?
