Let $K/k$ be a Galois extension with Galois group $\Gamma$ and let $X$ be a variety over $k$. Assume that either $X(k)\neq\varnothing$ or $\mathrm{Br}(k)=0$, the Brauer group of $k$. By the Hochschild-Serre spectral sequence, we have that $$\mathrm{Pic}(X)\cong \mathrm{Pic}(X_K)^{\Gamma}.$$

**Question 1**: Does the Galois action of $\Gamma$ preserve the numerical class of a line bundle on $X_K$? So that we can claim that $$N^1(X)=N^1(X_K)^{\Gamma}$$
where $N^1(X)$ is the $\mathbb{R}$-vector space of $\mathbb{R}$-Cartier divisors on $X$ modulo numerical equivalent.

If $X$ is an abelain or HypherKahler, then there is no problem. I want to know precisely how the Galois group $\Gamma$ act on $N^1(X_K)$.

**Question 2:** Does the Kawamata-Morrision conjecture about $X_K$ implies that of $X$?
The automorphism group $\mathrm{Aut}(X_K)$ acts on nef effective cone $A^e(X_K)$ has a fundamental domain $\Pi$ in the sense that

- $A^e(X_K)=\mathrm{Aut}(X_K)\cdot\Pi$;
- $\mathrm{Int}\space\Pi\cap g_*\mathrm{Int}\space\Pi=\varnothing$ unless $g_*=id$

I guess that $A^e(X)=A^e(X_K)^{\Gamma}$, and we can choose $\Pi^{\Gamma}$ to be the candidate of the fundamental domain.

Let $z\in A^e(X)$, regard it as an element in $A^e(X_K)$ we can find $g\in\mathrm{Aut}(X_K)$ such that $g_*z\in \Pi$. If $g\in\mathrm{Aut}(X)$, we are done, since then $\sigma\circ g\circ\sigma^{-1}=g$, and therefore $g_*z\in\Pi^{\Gamma}$. I don't know how to tackle the other case.