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$\DeclareMathOperator\Ob{Ob}$In Chan & Suen's paper A differential-geometric approach to deformation of pairs $(X, E)$ p.20, the authors define a Kuranishi map (or obstruction map) for the holomorphic pair $(X,E)$:$$\Ob_{(X,E)}:H^1(X,A(E))\to H^2(X,A(E)),$$ $$\sum_it_ie_i\mapsto\mathcal H[\phi(t),\phi(t)],$$ where $E$ is a holomorphic vector bundle over a compact complex manifold $X$ and $A(E)$ is the Atiyah extension of $E$.

If my understanding is right, the definition of the Kuranishi map for a compact complex manifold $X$ is \begin{gather*} \Ob_X:H^1(X,T_X)\to H^2(X,T_X), \\ \psi(t)\mapsto\mathcal H[\varphi(t),\varphi(t)], \end{gather*} where $\{e_1,\cdots,e_m\}$ is a base for $\mathbb H^1(X,T_X)\cong H^1(X,T_X)$, $\psi(t)=\sum\limits_{i=1}^me_it_i$, and $\varphi(t)$ is the solution of the following Kuranishi equation (see, for example, Kodaira & Morrow's book Complex manifolds p.161 Equation (10)) $$\varphi(t)=\psi(t)+\frac{1}{2}\bar\partial^*G[\varphi(t),\varphi(t)],$$ since this equation has a unique formal power series solution, the Kuranishi map $\Ob_X$ is well-defined. Recall that the solution $\varphi(t)$ satisfies the Maurer–Cartan equation if and only if $\mathcal H[\varphi(t),\varphi(t)]=0$. As a result, $X$ admits an unobstructed deformation if and only if the Kuranishi map $\Ob_X$ is a zero map.

If we let $E$ be a holomorphic line bundle $L$ over $X$, then the authors define a Kuranishi map (cf CS's paper p.23 Proposition 7.1) $$\Ob_L:H^1(X,\mathcal O)\to H^2(X,\mathcal O)$$ for the holomorphic line bundle $L$, (note that $Q=\operatorname{End}(L)$ and $H^1(X,Q)=H^1(X,\mathcal O)$).

My question is :

Does there exist a Maurer–Cartan equation for the deformations of holomorphic line bundle $L$ as in the complex manifold cases? If it exists, what is it? And is the space $\ker \Ob_L$ the deformation space of the holomorphic line bundle $L$?

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