Let $L \rightarrow X$ be an ample line bundle over $X$ 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 sence of Kodaira and Spencer? If one consider 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?