From first-order deformation to complex deformation of a pair $(X,L)$ $\DeclareMathOperator\Spec{Spec}$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?
 A: What you are typically looking for is a "true" deformation over an algebraic or analytic pointed curve $(T, 0)$ such that the tangent vector to $T$ at $0$ corresponds to the infinitesimal deformation you have.
The functor of infinitesimal deformations of a pair $(X, L)$ admits a semiuniversal formal deformation (see e.g. Sernesi's book page 146). Assuming $(X, L)$ is unobstructed (which is equivalent to saying that the base of the semiuniversal formal deformation is the Spec of a power series ring) then $H^1(X, \mathcal E_L)$ is the tangent space to the base of this deformation.
Now in order to construct $(T, 0)$:

*

*you can sometimes make an ad-hoc argument using some suitable Hilbert space, or


*you can use the extremely powerful Artin's approximation and algebraization theorems (see Sernesi's book pages 87, 88).
Artin's theorems ensures the existence of a "true" algebraic (I think there are also analytic versions of Artin's theorems) deformation provided the semiuniversal formal deformation is effective (see Sernesi's book page 82, for a Grothendieck's criterion for effectiveness).
This way you get an algebraic deformation over a smooth base, whose tangent space at a point $0$ is $H^1(X, \mathcal E_L)$, so you can get a path through $0$ pointing at your infinitesimal deformation.
