$\DeclareMathOperator\Spec{Spec}$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?