TITLE: Deformation theory over the field of algebraic numbers
QUESTION [11 upvotes]: Let $X_0$ be a smooth projective variety over $\mathbb{C}$ and let $\Theta_{X_0}$ be the locally free sheaf of $O_{X_0}$-module corresponding to tangent space of $X_0$. 
Goal: To find a sufficient condition on $X_0$ so that it admits a model over $\overline{\mathbb{Q}}$ (the field of algebraic numbers).
By spreading out $X_0$ we may choose a proper morphism 
$$
f:X\rightarrow Spec(\overline{\mathbb{Q}}[T_1,\ldots,T_n])=:B,
$$
where the $T_i$'s are "dependent variables" (i.e. they may satisfy some algebraic relations) such that when we specialize $T=(T_1,\ldots,T_n)$ to the point $P_0=(t_1,\ldots,t_n)\in\mathbb{C}^n$ we recover $X_0$. We may thus view 
$X$ via $f$ as a scheme over $Spec(\overline{\mathbb{Q}})$. Using sheaf cohomology, for every $\mathbb{C}$-valued point  $p$ of $B$, we get a connecting homomorphism
$$
\kappa:T_{B/Spec(\overline{\mathbb{Q}}),p}
\rightarrow H^1(X_p,\Theta_{X_p}).
$$
Note that an element $\partial\in T_{B/Spec(\overline{\mathbb{Q}}),p}$ may be viewed as a derivation of $\mathbb{C}$ over $\overline{\mathbb{Q}}$.
Now if we translate "naively" the Kodaira-Spencer deformation theory to our setting we should have a result which has the follwing flavor:
Tentative theorem: If for all $p\in B$ and all 
derivations $\partial\in T_{B/Spec(\overline{\mathbb{Q}}),p}$ one has that $\kappa(\partial)=0$ then $X_0$ admits a model over $\overline{\mathbb{Q}}$.
Question: Do we have such a result and if the answer is yes then what is a good reference where it is proved?
I would like a reference where the proof is as transparent as possible.

REPLY [2 votes]: The answer is yes, with the modification that the descent may not be from $B$ to $\overline{\mathbb Q}$ but from some etale cover of $B$.  I.e. it really is a result about algebraically closed fields of definition. (The intuition being that Kodaira-Spencer trivial implies that a family is isotrivial).
The following was first proved by Buium:
Theorem Let $X$ be a variety, proper over an algebraically closed ﬁeld $K$.
Then $X$ is deﬁned over the ﬁxed ﬁeld of the set of all derivations of $K$ which lift to
derivations of the structure sheaf of $X$.
See: 
Buium, Alexandru; Diﬀerential function ﬁelds and moduli of algebraic varieties. Lecture Notes n Mathematics, 1226. Springer-Verlag, Berlin, 1986.
Buium, Alexandru; Fields of deﬁnition of algebraic varieties in characteristic zero. Compositio Math. 61 (1987), no. 3, 339–352.
and also:
Gillet,  Henri; "Differential algebra - A Scheme Theory Approach", in Differential algebra and related topics: proceedings of the International Workshop, Newark Campus of Rutgers, The State University of New Jersey, 2-3 November 2000, Editors    Li Guo, William F. Keigher, World Scientific
The converse statement is of course trivial