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.