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.
Best Answer
I think your question has a positive answer: The Kodaira-Spencer maps at each point $p \in B$ fit together to give a map of sheaves $\Theta_B \to R^1 f_* \Theta_{X/B}$ and your condition implies that this is map is zero. One may then base change to $C$ and apply Kuranishi's theorem (On the locally complete families of complex analytic structures. Ann. of Math. (2) 75 1962 536–577, in particular, Theorem 3) to deduce that all fibres are isomorphic as complex manifolds and hence, by GAGA, also as algebraic varieties. (The point is that one does not need a global moduli space, it suffices to have something that works analytically locally and this is supplied by Kuranishi's theorem.)
It is possible to give a purely algebraic proof of this by replacing the Kuranishi space by the Hilbert scheme of closed subschemes of $P^n$ for some large $n$ (depending on $X$) and analyzing the tangent map of the map from $B$ to such a scheme induced by choosing a projective embedding of $X$. This requires some more arguments since distinct points in the Hilbert scheme do not necessarily correspond to non-isomorphic subvarieties.