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(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?

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    $\begingroup$ Zero-dimensional complex-analytic spaces are the same as 0-dimensional schemes locally of finite type over $\mathbf{C}$. To make good use of infinitesimal techniques in complex-analytic geometry one must abandon manifolds for the broader framework of complex-analytic spaces (which admits good notions of smooth, etale, and flat morphisms with many familiar properties -- flatness via stalks as locally ringed spaces -- but with harder proofs than for schemes). Read the beautiful book "Coherent Analytic Sheaves" for first steps, and then "Algebraic methods in the global theory of complex spaces". $\endgroup$
    – nfdc23
    Oct 1, 2016 at 22:24

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This was pretty much answered by nfdc23 in a comment, but let me make this official. To makes sense of this, you need to work in the category of analytic spaces, rather than just complex manifolds. Very roughly, analytic spaces are to manifolds, what schemes are to varieties. There is functor which takes a scheme $X$ locally of finite type over $\mathbb{C}$ to an analytic space $X^{an}$ (see SGA1 exp XII, although this is more than you need). Replace everything in the diagram by $Spec A^{an}$ etc. Now the diagram makes sense. Once you realize that's what going on, feel free to drop it in the future.

If $X\to D$ is a deformation over a polydisk as in Kodaira, you can take the fibre product with a morphism $Spec A^{an}\to D$ to get an infinitesimal deformation.

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