$\DeclareMathOperator\Spec{Spec}$Let $L \rightarrow X$ be an ample line bundle over $X$ which is a compact complex manifold. Suppose that I have a ***first-order*** deformation of the pair $(X,L)$. When does this first-order deformation gives rise to a **complex** deformation of the pair $(X,L)$ in the sense of Kodaira and Spencer? 
If one considers the extension 
$$ 0 \rightarrow \mathcal{O}_X \rightarrow \mathcal{E}_L \rightarrow T_X \rightarrow 0$$
defined by the first Chern class of $L$,
then $H^2(X, \mathcal{E}_L)$ is an obstruction space for the functor of **infinitesimal deformations** of $(X,L)$. Is $H^2(X, \mathcal{E}_L)=0$ sufficient to ensure the existence of an associated **complex** deformation? 

 I took the definitions from Sernesi's book. I call a deformation of $X$ a flat surjective morphism $$\mathcal{X} \rightarrow \Delta$$ with $X \rightarrow \Spec(\mathbb{C})$ the central fiber. Then the deformation is:

- "**first-order**" if $\Delta=\Spec(\mathbb{C}[\epsilon])$

- "**infinitesimal**" if $\Delta=\Spec(A)$ with $A$ a local artinian $\mathbb{C}$-algebra.

- "**complex**" if $\Delta$ is a complex manifold. 

In that case the definition of the morphism $$\mathcal{X} \rightarrow \Delta$$ is to be a proper submersion.

The point that I do not understand is how do I go from infinitesimal to complex?