(Note: originally asked on stackexchange)

Let $X$ be a compact, complex manifold and $A\in \textbf{Art}$ where $\textbf{Art}$ is the category of local artinian $\mathbb{C}$-algebras with residue field $\mathbb{C}$.

In page 24 of this paper they define an infinitesimal deformation of $X$ over $Spec(A)$ to be the following diagram:

$$ \matrix{ X & \overset{i}{\rightarrow} & X_A \cr \downarrow & & \downarrow \pi \cr Spec(\mathbb{C}) & \overset{i}{\rightarrow} & Spec(A) \cr } $$

where $\pi$ is proper and flat holomorphic map, $a\in Spec(A)$ is the closed point, $i$ is a closed embedding, and $X\simeq X_A\times_{Spec_(A)} Spec(\mathbb{C})$.

This definition looks a lot like a deformation of a scheme. But $X,X_A$ are not necessarily schemes (or are they?).

My question is, does this definition make sense? What are we considering $Spec(A)$ as? And what is a flat morphism of manifolds?

Also, in ch. 4 of his book, Kodaira defines a deformation of a complex manifold $X$ to be a deformation of the transitions functions that glue together the polydisks that cover $X$. Are these definitions the same?